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\begin{document}

\title[A Kreweras map on positive $m$-divisible non-crossing partitions]{A Kreweras map on positive $m$-divisible non-crossing partitions for real reflection groups}

\author[C.~Krattenthaler]{Christian Krattenthaler$^*$}
\address[C.~Krattenthaler]{Fakult\"at f\"ur Mathematik, Universit\"at Wien, 
Oskar-Morgenstern-\break Platz~1, A-1090 Vienna, Austria.}
%\email{}
\urladdr{http://www.mat.univie.ac.at/\lower0.5ex\hbox{\~{}}kratt}
\thanks{$^*$Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S50-N15,
the latter in the framework of the Special Research Program
``Algorithmic and Enumerative Combinatorics"}

\author[C.~Stump]{Christian Stump$^\dagger$}
\address[C.~Stump]{Institut f\"ur Mathematik, Freie Universit\"at Berlin, Germany}
\email{christian.stump@fu-berlin.de}
\urladdr{http://homepage.univie.ac.at/christian.stump/}
\thanks{$^\dagger$Research supported by the German Research Foundation DFG, grant STU 563/2-1 "Coxeter-Catalan combinatorics"}

\subjclass [2010]{Primary 05E15;
Secondary 05A05 05A10 05A15 05A18 06A07 20F55}
\keywords {root systems, reflection groups, Coxeter groups,
generalised non-crossing partitions, annular non-crossing partitions,
chain enumeration, M\"obius function, 
Chu--Vandermonde summation}

\begin{abstract}
\end{abstract}

\maketitle

%\setcounter{tocdepth}{1}
% \tableofcontents

\section{Introduction}
\label{sec:0} 

Non-crossing partitions have been introduced into combinatorial
mathematics by Kreweras \cite{KrewAA} in 1972.
Biane \cite{BianAA} presented an algebraic realisation in terms
of certain factorisations of a long cycle in the symmetric
group. These classical non-crossing partitions were lifted
to all finite Coxeter groups and a given Coxeter element by Bessis \cite{BesDAA},
and independently by Brady and Watt \cite{BRWaAB}.
The earlier combinatorial construction
of non-crossing partitions for type~$B$ due to Reiner
\cite{ReivAG} turned out to the equivalent to the
type-$B$ case of the construction of Bessis, Brady and Watt,
while Reiner's construction in the same paper for type~$D$
did not fit into the general framework of \cite{BesDAA,BRWaAB}.
Inspired by a refinement of Kreweras' classical non-crossing partitions
due to Edelman \cite{EdelAA}, Armstrong \cite{Arm2006} introduced
a positive integer parameter~$m$ and defined ``$m$-divisible
non-crossing partitions" for any finite Coxeter group~$W$ and a Coxeter element $c \in W$,
together with a natural partial order.
Armstrong showed that, in type~$A_{n-1}$ with~$c = (1,2,\ldots,n)$, these objects can be
pictorially realised as non-crossing partitions of
$\{1,2,\dots,mn\}$ in the sense of Kreweras where all
block sizes are divisible by~$m$ (thus recovering the
$m$-divisible non-crossing partitions of Edelman~\cite{EdelAA}),
while, in type~$B_n$ with~$c = (1,2,\ldots,n,-1,-2,\ldots,-n)$, they can be pictorially realised
as non-crossing partitions of
$\{1,2,\dots,mn,-1,-2,\dots,-mn\}$ which remain invariant
under the simultaneously replacement $i\mapsto-i$, $i=1,2,\dots,mn$,
and where again all block sizes are divisible by~$m$.
The first author and M\"uller \cite{KrMuAB} presented a pictorial 
realisation of $m$-divisible non-crossing partitions of type~$D_n$
with~$c = \big(1,2,\ldots,n-1,-1,-2,\ldots,-(n-1)\big)\big(n,-n\big)$
as (classical) non-crossing partitions of 
$\{1,2,\dots,mn,-1,-2,\dots,-mn\}$ which are arranged on
an annulus subject to similar, but more
technical conditions. In all these pictorial realisations,
the above mentioned partial order corresponds to 
the classical refinement order on set partitions.

As observed in various places (see, for example,
\cite[Prop.~9]{Cha2004}),
one of the remarkable facts about these generalised non-crossing
partitions is that their total number is given by the
uniform formula 
\begin{equation}
%   \big| \mNC \big| =
  \mCat(W) :=
  \prod_{i=1}^n{\frac{e_i+mh+1}{e_i+1}}, 
\label{eq:mdivnc} 
\end{equation}
where $d_1 \leq \ldots \leq d_n$ denote the \defn{degrees of the fundamental
invariants} for~$W$ and where~$h$ is the \defn{Coxeter
  number} (see \Cref{sec:prel} for precise definitions.)
These numbers are nowadays called the
\defn{Fu\ss-Catalan numbers} for finite Coxeter groups.
They also count various other combinatorial objects
associated with~$W$. Among these, there are
for example the generalised clusters of 
Fomin and Reading \cite{FR2005}. 

In the present article, we study a subset of the $m$-divisible
non-crossing partitions associated with~$W$, namely
\defn{positive} $m$-divisible non-crossing partitions; see Definition~\ref{def:NCpos}. 
For $m=1$, these have been introduced
by Reading \cite[Cor.~9.2 plus the following paragraph]{Rea2007}
(implicitly) and by Sim\~oes \cite[Thm.~1.1]{Sim2012} 
(explicitly; only for crystallographic types). 
For generic~$m$, positive
$m$-divisible non-crossing partitions appear in~\cite[Cor.~7.4 plus the following
  paragraph]{BRT2012}
%CK: Wenn man ``article'' weglaeszt, dann kann man nicht sagen
%   ``by Buan, Reiten and Thomas''.
(again, only for crystallographic types), and in~\cite{STW2015}.
%CK: ebenso.
% by the second author, Thomas and Williams.

The total number of positive $m$-divisible non-crossing
partitions is given by
the \defn{positive Fu\ss-Catalan number} (see~\cite[Cor.~7.4]{BRT2012}
for crystallographic types and \cite[Ch.~6]{STW2015} for all types)
\begin{align}
\big| \mNCplus \big| = \mCatplus(W) :=
\prod_{i=1}^\ell{\frac{e_i+mh-1}{e_i+1}}. 
\label{eq:mdivncplus} 
\end{align}
Here, the term ``positive" refers to the fact that the same
numbers also count ``positive generalised clusters" in the
sense of \cite{FR2005}. Indeed, both Buan, Reiten and Thomas, and the second author, Thomas and Williams used
\cite[Cor.~12.2]{FR2005} to derive the above counting formula
for positive $m$-divisible non-crossing partitions.

\medskip

Our first focus is on describing positive $m$-divisible non-crossing
partitions for the classical types $A_{n-1}$, $B_n$, and $D_n$ as subsets
of the above mentioned pictorial realisations.
As we show in \Cref{prop:1}, in type~$A_{n-1}$
the condition of being positive corresponds to the condition
that the block containing~$1$ also contains~$mn$.
\Cref{prop:1B} says that being positive in type~$B_n$ means
that the block containing~$1$ also contains a negative element,
and \Cref{prop:1D} says that being 
positive in type~$D_n$ means
that the block containing~$1$ also contains an element
from $\{-1,-2,\dots,-m(n-1)\}$.

\medskip

Our next set of results consists of counting formulas for
types~$A$, $B$, and~$D$ that refine the total count
\eqref{eq:mdivncplus}. More precisely, these formulas count 
positive $m$-divisible non-crossing partitions, and chains
thereof, with restrictions on the block structure and
the ranks of elements in the chains, see \Cref{??}.

\medskip
Our final set of results addresses a cyclic sieving phenomenon
for the positive $m$-divisible non-crossing partitions associated
with a finite Coxeter group~$W$. The phenomenon of cyclic sieving
has been introduced by Reiner, Stanton and White in \cite{RSW2004}.
It refers to a certain striking enumerative property of
combinatorial objects under the action of a cyclic group.
On (ordinary) $m$-divisible non-crossing partitions associated
with~$W$, there exists a natural action of the cyclic group
due to Armstrong \cite[Def.~3.4.15]{Arm2006}, which can be considered
as an extension of an action on classical non-crossing partitions
due to Kreweras, which is now known as the ``Kreweras complement". 
Armstrong's map translates
into ordinary rotation on the pictorial realisations of
$m$-divisible non-crossing partitions in types~$A$, $B$, and~$D$.
Armstrong \cite[Conj.~5.4.7]{Arm2006} went
on to conjecture a cyclic sieving phenomenon for this action.
A related conjecture on cyclic sieving of $m$-divisible
non-crossing partitions associated with~$W$ was made by
Bessis and Reiner~\cite[Conj.~6.5]{BR2007}. Both conjectures
were proved by the first author and M\"uller~\cite{KratCG,KrMuAD}.

Inspired by a representation-theoretic functor in the work of
Sim\~oes \cite{Sim2012} and by Buan, Reiten and Thomas
\cite{BRT2012}, we introduce 
a ``positive analogue'' of the (generalised) Kreweras complement on 
positive $m$-divisible non-crossing 
partitions in \Cref{def:positivekreweras}.
In types~$A$, $B$, and $D$, it translates into ``pseudo-rotations,"
meaning that it can be described by a ``deformation" of rotation;
see \Cref{prop:2,prop:2B,prop:2D}.
We then prove that the positive $m$-divisible non-crossing partitions
associated with~$W$ together with the positive Kreweras map satisfy a
cyclic sieving phenomenon,
see \Cref{thm:3,??}
??refined cyclic sieving/blocks;
conjecture cyclic sieving for fixed type??.

\medskip

Our article is organised as follows.
In \Cref{sec:prel}, we recall the definitions of
$m$-divisible non-crossing partitions associated with a
Coxeter group~$W$, and their Kreweras complements. 
There, we also define positive $m$-divisible non-crossing
partitions and the positive analogue of the Kreweras complement.
Moreover, we collect several properties of these objects
(see \Cref{thm:positivedescription} and \Cref{cor:unique}),
and we state two of the main results, namely the cyclic sieving
phenomena in \Cref{thm:CS}.
% loeschen??In \Cref{sec:representation theory}, we recall an
% alternative description of positive
% $m$-divisible non-crossing partitions and the positive
% Kreweras complement in terms of subwords
% of an infinite repetition of a reduced word for the Coxeter element.
% This description is then used to provide a proof of \Cref{thm:order}
% on the order of the positive Kreweras complement.
This is followed by \Cref{sec:typeA,sec:typeB,sec:typeD}, in 
which the positive $m$-divisible non-crossing partitions and
their positive Kreweras complement are discussed for the classical
types. Each of these sections, starts with a translation of
the algebraic definition of these objects into the language
of set partitions; see ??. Subsequently, a subsection is devoted
to the enumeration of (chains?? of) positive $m$-divisible non-crossing
partitions in each type subject to various restrictions on
their block structure, see \Cref{sec:enumA,sec:enumB,sec:enumD}. 
In a further subsection, we characterise, for each type, 
the positive $m$-divisible non-crossing partitions that are invariant
under a given power of the positive Kreweras complement. 
Furthermore, we enumerate
the invariant those and use this the obtained formulae to deduce
the corresponding cyclic sieving phenomenon, see
\Cref{sec:rotA,sec:rotB,sec:rotD}.
In \Cref{sec:typeExc}, we finally treat the exceptional types
with respect to invariance and cyclic sieving under the action
of the positive Kreweras complement. We close our article
with \Cref{sec:Panyushev}, where we revisit 
two conjectures of Panyushev \cite{Pan2008}
on antichains in root posets and discuss enumerative similarities
with the orbit sizes of the positive Kreweras complement.

\section{Preliminaries and main results}
\label{sec:prel}

In this section, we provide the necessary background on finite Coxeter
groups and non-crossing partitions.
For more background and detailed treatments we refer
to~\cite{Arm2006,STW2015}.
We then define the main objects of study and present the 
%CK: major
main results.

\medskip

Let $(W,\reflS)$ denote a finite Coxeter system of rank~$\rk(W) = n$ with simple roots $\Delta = \{ \alpha_s : s \in \reflS \}$ and root system $\Phi = \big\{ w(\alpha_s) : w \in W, s \in \reflS \big\} = \Phi^+ \sqcup \Phi^-$.
A \defn{standard parabolic subgroup} of~$(W,\reflS)$ is a Coxeter subsystem~$(W',\reflS')$ generated by a subset $\reflS' \subseteq \reflS$.
Next, denote by
$$\reflR = \{ wsw^{-1} : w \in W, s \in \reflS\} = \{ s_\beta : \beta \in \Phi^+ \}$$
the set of \defn{reflections} in~$W$ and 
%CK: set
let $N = |\reflR|$.
Given $w\in W$, we let $\lenR(w)$ be the \defn{reflection length} of
$w$, which is the length of a reduced $\reflR$-word
for~$w$. Analogously to the weak order on~$W$, one defines the
\defn{reflection order}~$\leqR$ by letting $u\leqR v$ if 
$\lenR(u)+\lenR(u^{-1}v)=\lenR(v)$. Thus, $u\leqR v$ if
and only if there is a reduced $\reflR$-word for~$v$ which begins
with a reduced $\reflR$-word for~$u$.
A \defn{(standard) Coxeter element} $c$ is given by the product of all generators~$\reflS$ 
%CK: is
in any order, $c = \prod_{s \in \reflS} s \in W$.
Observe that all standard Coxeter elements are conjugate to each other and that their order is given by the \defn{Coxeter number}~$h$, which moreover 
%CK: satisfies the relation
is given by $N = nh/2$.

For a Coxeter element~$c$, one defines the set of
\defn{non-crossing partitions} $\NC$ as the set of all elements
$w\in W$ with $w\leqR c$. This set together with the partial
order $\leqR$ forms a lattice (cf.~\cite{BRWaAB}). 
Since $\leqR$ is invariant under
conjugation, this lattice is independent of the choice of~$c$.

Armstrong's set of
\defn{$m$-divisible non-crossing partitions}~\cite{Arm2006} is defined by
\begin{multline*}
\mNC=\big\{(w_0,w_1,\dots,w_m):w_0w_1\cdots w_m=c\text{ and }\\
\ell_\reflR(w_0)+\ell_\reflR(w_1)+\dots+\ell_\reflR(w_m)=n \big\}.
\end{multline*}
For $m=1$, this definition reduces to the definition of 
non-crossing partitions $\NC$, by identifying $(w_0,w_1)$
with $w_0$.
The set $\mNC$ becomes a partially ordered set by letting
$$
(u_0,u_1,\dots,u_m)\leqR
(v_0,v_1,\dots,v_m)\quad \text{if }
u_i\geqR v_i,\text{ for }i=1,2,\dots,m.
$$
(We choose this convention so that, in classical types, this
partial order corresponds to the refinement order on set
partitions; see \Cref{??}.)
As a partially ordered set, $\mNC$ does again not depend on the
choice of the Coxeter element~$c$. In general, it is not a
lattice but only a join-semilattice. 
Observe that for a given $m$-divisible non-crossing partition $(w_0,w_1,\dots,w_m)$, the elements $w_0 \cdots w_{m-1}$ and $w_1 \cdots w_m$ are both contained in $\NC$.
In particular, note here that choosing $w_0$ or $w_m$ as being special is a matter of choice.
We indeed need to ``mix'' the two choices in \Cref{def:NCpos,def:positivekreweras} as this will make it possible to obtain simple counting formulae according to type (or ``block structure'') in classical types.

\medskip

The main object of study in this paper is the following subset of $m$-divisible non-crossing partitions.

\begin{definition} \label{def:NCpos}
  An $m$-divisible non-crossing partition $(w_0,w_1,\dots,w_m)$ is called \defn{positive} if the product $w_0 \cdots w_{m-1}$ does not live in any proper standard parabolic subgroup of~$W$.
  The set of all \defn{positive $m$-divisible non-crossing partitions} in $\mNC$ is denoted by $\mNCplus$.
\end{definition}
This definition is motivated by a bijective correspondence between $m$-divisible non-crossing partitions and the generalized cluster complex defined by Fomin and Reading~\cite{FR2005}.
In~\cite{BRT2012}, Buan, Reiten, and Thomas provided a representation theoretic bijection between facets in the $m$-generalized cluster complex and $m$-divisible non-crossing partitions for crystallographic types.
This bijection was made combinatorial and uniform for all types by the second author, Thomas and Williams in~\cite{STW2015}.
Under this bijection, positive $m$-divisible non-crossing partitions 
correspond exactly to \lq\lq positive clusters\rq\rq\ which are counted, see~\cite{FR2005}, by the \defn{positive Fu\ss-Catalan numbers}
\eqref{eq:mdivncplus}.
In~\cite{KratCG}, the first author provides more refined versions of the counting formula~\eqref{eq:mdivnc} for $m$-divisible non-crossing partitions according to the rank in the poset of $m$-divisible non-crossing partitions, and as well according to block sizes in a combinatorial model in the classical types.
In \Cref{sec:typeA,sec:typeB,sec:typeD}, we provide similar refinements for the positive $m$-divisible non-crossing partitions.

\medskip

Beside refined counting formulae, we study in this paper a ``positive analogue'' of the \defn{Kreweras map} on $m$-divisible non-crossing partitions.
This map and a variant thereof were introduced by Armstrong in~\cite[Definition~3.4.15]{Arm2006}.
These are given by
\begin{align*}
   \Krew_m &:
  \begin{array}{ccc}
     \mNC &\tilde\longrightarrow& \mNC \\
     (w_0,w_1,\ldots,w_m) &\mapsto& (cw_mc^{-1},w_0\ldots,w_{m-1}),
  \end{array}\\[5pt]
   \Krewtilde_{m+1} &:
  \begin{array}{ccc}
     N\!C^{(m+1)}(W,c) &\tilde\longrightarrow& N\!C^{(m+1)}(W,c) \\
     (w_*,w_0,w_1,\ldots,w_m) &\mapsto& (v,cw_mc^{-1},w_0\ldots,w_{m-1}),
  \end{array}
\end{align*}
where $v = (cw_mc^{-1})w_*(cw_mc^{-1})^{-1}$.
Both maps are close relatives, the embedding of $\mNC$ into $N\!C^{(m+1)}(W,c)$ given by
$$
  \ph :
  \begin{array}{ccc}
    \mNC &\rightarrow& N\!C^{(m+1)}(W,c) \\
    (w_0,w_1,\ldots,w_m) &\mapsto& (\one,w_0,w_1,\ldots,w_m)
  \end{array}
$$
has the property that it maps $\Krew_m$ to $\Krewtilde_{m+1}$. In symbols, $\ph \circ \Krew_m = \Krewtilde_{m+1} \circ \ph$.
Moreover, $\Krewtilde_1 = \Krew_1 \circ \Krew_1$.
We drop the index in $\Krew_m$ and in $\Krewtilde_{m+1}$ if it is clear from the context and only write $\Krew$ and $\Krewtilde$, respectively.

\bigskip

The Kreweras maps $\Krew$ and $\widetilde\Krew$ do not stabilize the set of positive $m$-divisible non-crossing partitions.
The main motivation for this paper is to define and study positive versions of both maps.
For this purpose, we need to provide another, slightly more involved way to describe (positive) $m$-divisible non-crossing partitions.
This approach is based on a certain reduced $\reflR$-word for $w \in \NC$ depending on the Coxeter element~$c$.

\medskip

For this construction, we need to recall some background on Coxeter-sortable elements in Coxeter groups as introduced by Reading in~\cite{Rea2005} and further studied in various places, see for example~\cite{RS2008,CLS2011}.
Fix a reduced word $\c = c_1 \cdots c_n$ for the Coxeter element~$c$, and define the \defn{$\c$-sorting word} $\w(\c)$ for $w \in W$ to be the lexicographically first reduced word for~$w$ when written as a subword of $\c^\infty$.
Observe that all reduced words for~$c$ are obtained from each other by transpositions of consecutive commuting letters, or \defn{up to commutations} for short.
Thus, given the Coxeter element~$c$, the various $\c$-sorting words for $w \in W$ all coincide up to commutations.
Let furthermore $\wwo(\c) = s_{k_1} \cdots s_{k_N}$ be the $\c$-sorting word of the longest element $\wo \in W$.
It was shown for example in~\cite[Proposition~7.1]{CLS2011} that, after a suitable interchanging of consecutive commuting letters, $s_{k_{N-n+1}} \dots s_{k_N} = \psi(\c) = \psi(c_1) \dots \psi(c_n)$, where $\psi$ is the involution on $\reflS$ given by $\psi:s \mapsto \wo s \wo$.
Depending on the given Coxeter element~$c$, this word $\wwo(\c)$ induces a \defn{reflection ordering} on $\reflR$ given by
\begin{equation} \label{eq:refls} 
\big\{ s_{k_1} \lec s_{k_1} s_{k_2} s_{k_1} \lec
  s_{k_1} s_{k_2} s_{k_3} s_{k_2} s_{k_1} \lec \dots \lec
  s_{k_1} \dots s_{k_{N-1}} s_{k_N} s_{k_{N-1}} \dots s_{k_1} \big\}.
\end{equation}
For a definition (which we do not need in this paper) and further properties of reflection orderings, see \cite{Dye1993}.
For later reference, 
%CK: Ich verstehe den folgenden Satz aus mehreren Gruenden nicht.
%   Erstens verstehe ich ``denote'' hier nicht. ``Denote'' bedeutet
%   ``bezeichnen''. Wie kann man etwas mit einem *Wort* bezeichnen?
%   Zweitens, warum ist $\invc$ ein Wort?
%   Drittens, was ist die Inversionsfolge eines Wortes?
we denote this reflection ordering by the 
word~$\invc$ as this is indeed the inversion sequence of the word 
$\Rword(\wo)$.
%CK: Der naechste Satz sieht so aus, als ob er lange immer wieder
%    umformuliert wurde, bis er kein richtiger Satz mehr war.
%   Weiters: Was bedeutet "removed by $\invc^L$"?
We similarly denote this word with the final~$n$ letters corresponding to $\psi(c)$ removed by $\invc^L$, and these final~$n$ letters by $\invc^R$, so that $\invc = \invc^L \invc^R$.
Now, associate to every element $w \in \NC$ a reduced $\reflR$-word of~$w$ given by the unique subword $r_1 \cdots r_\ell$ of $\invc$ that is a reduced $\reflR$-word for~$w$.
In other words, $r_1 \cdots r_\ell$ is the unique reduced $\reflR$-word for~$w$ such that $r_1 \lec r_2 \lec \cdots \lec r_\ell$, see~\cite[Remark~6.8]{Rea2007}.
%CK: In jedem Fall sieht es fuer mich auch so aus, dasz diese technischen
%    Definitionen mit einem Beispiel illustriert werden muessen, sonst
%    kennt man sich ueberhaupt nicht aus.
Indeed, this factorisation is the unique factorisation of~$w$ into reflections induced by the EL-labelling of the non-crossing partition lattice in~\cite{ABW2007}.

\medskip

This unique factorisation of non-crossing partitions yields the following alternative two descriptions of $m$-divisible non-crossing partitions, see~\cite[???]{STW2015}.

\begin{proposition}\label{prop:worddescription}
  Let~$c$ be a Coxeter element, and let $(w_0,w_1,\ldots,w_m) \in \mNC$ be an $m$-divisible non-crossing partition.
  The map sending $(w_0,w_1,\ldots,w_m)$ to
  \begin{enumerate}
    \item the subword of $\invc^{m+1}$ where $w_i$ is mapped to its unique subword inside the $(i+1)$\st\ copy of $\invc$ for $0 \leq i \leq m$ yields a bijection to all subwords of $\invc^{m+1}$ whose product is~$c$, and
    \item the subword of $\invc^{m}$ where $w_i$ is mapped to its unique
%CK: subword
subword inside the $i$\th\ copy of $\invc$ for $1 \leq i \leq m$ yields a bijection to all subwords of $\invc^{m}$ whose 
%CK: produce
product is in $\NC$.
  \end{enumerate}
\end{proposition}

This has another enumerative implication,
%CK: given by
namely a formula for the Fu\ss-Catalan numbers in terms of 
%CK: sums of binomials.
a sum of binomial coefficients.

\begin{corollary}\label{cor:fusscatcounting}
  \begin{equation}
    \mCat(W) = \sum \binom{m+1+\asc(t_1,\ldots,t_n)}{n},
  \end{equation}
  where the sum ranges over all factorisations 
%CK: $t_1,\ldots,t_n$ of the Coxeter element $c = t_1 \cdots t_n$,
$c = t_1 \cdots t_n$ of the Coxeter element into reflections $t_i$,
for $i=1,2,\dots,n$,
and where $\asc(t_1,\ldots,t_n)$ is the number of indices~$i$ such that $t_i <_c t_{i+1}$.
\end{corollary}
\begin{proof}
  Given a factorisation $t_1,\ldots,t_n$ of~$c$, the binomial in the sum counts the number of ways to place the~$n$ letters $t_1,\ldots,t_n$ into their (unique) positions inside the $m+1$ copies of $\invc$ such that two consecutive letters $t_i$ and $t_{i+1}$ are allowed in the same copy if and only if $t_i <_c t_{i+1}$.
  In other words, this counts the number of subwords of $\invc^{m+1}$ that are reduced $\reflR$-words for~$c$ and where the reflections are $t_1,\ldots,t_n$ in this order.
  Summing over all possible factorisations yields the statement.
\end{proof}

Moreover, this unique factorisation can be used to determine positive $m$-divisible non-crossing partitions as follows.
Denote by
\begin{equation} \label{eq:LR} 
  w = w^L w^R
\end{equation}
the decomposition of~$w \in W$ such that $w^L$ and $w^R$ are the parts of the unique subword for~$w$ of $\invc$ that lie inside $\invc^L$ and inside $\invc^R$, respectively.

\medskip

The following theorem gives an alternative description of positive
$m$-divisible non-crossing partitions. 
It was first observed in crystallographic types
in~\cite[Thm.~7]{BRT2012}. The general case was established
in~\cite[Ch.~6]{STW2015}.

\begin{theorem}
\label{thm:positivedescription}
  $(w_0,w_1,\dots,w_m) \in \mNC$ is positive if and only if $w_m = w_m^L$.
\end{theorem}

We note that this theorem has
the following immediate consequences.

\begin{corollary} \label{cor:unique}
  Let $w \in \NC$, and decompose~$w$ in the form $w=uv$ such that
  \begin{enumerate}
    \item $\lenR(u)+\lenR(v)=\lenR(w)$;
    \item $v=r_1\cdots r_k$, with $r_i$ one of the last $n$ reflections
          in~\eqref{eq:refls}, for $i=1,2,\dots,k$, and
          $r_1\lec\dots\lec r_k$; 
    \item $cu^{-1}$ does not live in a standard parabolic subgroup
      of~$W$. 
  \end{enumerate}
  Then $w^L=u$ and $w^R=v$.
\end{corollary}

\begin{proof}
  Since $cu^{-1}$ does not live in a standard parabolic subgroup,
  we have $(cu^{-1},u)\in N\!C_+^{(1)}(W,c)$ by
  \Cref{def:NCpos}. 
  \Cref{thm:positivedescription} then implies that
  $u^L=u$. By definition, $u^L$ is formed by determining the
  subword of $\reflR$ that is a reduced
  $\reflR$-word for $u$, and then disregarding those reflections
%CK: which lie
  which lie inside $\invc^R$.
  Since $u^L=u$, nothing had to be disregarded.
  We conclude by observing that $w^L=(uv)^L = u^L = u$ and $w^R = (uv)^R = u^Rv = v$.
\end{proof}

\begin{proposition} \label{prop:positiveworddescription}
  The bijections in \Cref{prop:worddescription} restrict to bijections between positive $m$-divisible non-crossing partitions $\mNCplus$ and
  \begin{enumerate}
    \item subwords of $\invc^{m}\invc^L$ whose product is~$c$, and
    \item subwords of $\invc^{m-1}\invc^L$ whose 
%CK: produce
product is in $\NC$.
  \end{enumerate}
\end{proposition}

We again have the following corollary, with the same proof as \Cref{cor:fusscatcounting}.

\begin{corollary}\label{cor:positivefusscatcounting}
%CK: We have
We have
  \begin{equation}
    \mCatplus(W) = \sum \binom{m + \asc(t_1,\ldots,t_n,t)}{n},
  \end{equation}
  where the sum ranges over all factorisations 
%CK: $t_1,\ldots,t_n$ of the Coxeter element $c = t_1 \cdots t_n$, 
$c = t_1 \cdots t_n$ of the Coxeter element into reflections $t_i$,
for $i=1,2,\dots,n$,
and where~$t$ is the first reflection in the word $\invc^R$.
%CK: Stimmt das auch?
%    Duerfen wir *beliebige* Faktorisierungen $c = t_1 \cdots t_n$
%    nehmen?
%    Was ist die Rolle von t hier?
%    Es gilt ja dann $c = t_1 \cdots t_n t = c t$. Soll das so sein?
\end{corollary}

Using these descriptions of positive $m$-divisible non-crossing partitions, we can now give the main definition of this paper.

\begin{definition}
\label{def:positivekreweras}
  The \defn{positive Kreweras maps} $\Krewplus$ and $\Krewplustilde$ are given by
  \begin{align*}
    (\Krewplus)_m &:
      \begin{array}{ccc}
        \mNCplus &\tilde\longrightarrow& \mNCplus \\
        (w_0,w_1,\ldots,w_m) &\mapsto& (cw_{m-1}^Rw_mc^{-1},w_0,w_1,\dots,w_{m-1}^L),
      \end{array} \\[5pt]
    (\Krewplustilde)_{m+1} &:
      \begin{array}{ccc}
        N\!C^{(m+1)}_+(W,c) &\tilde\longrightarrow& N\!C^{(m+1)}_+(W,c) \\
        (w_*,w_0,w_1,\ldots,w_m) &\mapsto& (v_+,cw_{m-1}^Rw_mc^{-1},w_0,w_1,\dots,w_{m-1}^L),
      \end{array}
  \end{align*}
  where $v_+ = (cw_{m-1}^Rw_mc^{-1})w_*(cw_{m-1}^Rw_mc^{-1})^{-1}$.
\end{definition}
The embedding $\ph$ of $\mNC$ into $N\!C^{(m+1)}(W,c)$ given above also embeds $\mNCplus$ into $N\!C^{(m+1)}_+(W,c)$, and we again have that $\ph \circ (\Krewplus)_m = (\Krewplustilde)_{m+1} \circ \ph$.
We again usually drop the indices in $(\Krewplus)_m$ and in $(\Krewplustilde)_{m+1}$ and write $\Krewplus$ and $\Krewplustilde$, respectively.

\medskip

The two maps $\Krewplus$ and $\Krewplustilde$ can as well be described in terms of words as in \Cref{prop:positiveworddescription}.
%CK: To this, 
In order to do so, one considers the words $\invc^{m}\invc^L$ and $\invc^{m-1}\invc^L$ in \Cref{prop:positiveworddescription} as cyclic words where the final missing copy of $\invc^R$ is identified with the initial~$n$ letters in the initial copy of $\invc$.
% Observe importantly that this cyclic word is independent of the chosen Coxeter element.

\begin{corollary}\label{cor:poskrewwords}
  Under the bijections above and identifying letters in the word as described, $\Krewplus = (\Krewplus)_m$ and $\Krewplustilde = (\Krewplustilde)_{m+1}$ both correspond to sending a letter in the $i$\th\ copy of $\invc$ inside $\invc^{m}\invc^L$ to the same letter in the $(i+1)$\st\ copy for $1 \leq i \leq m+1$.
\end{corollary}

Using this description, we first obtain that $\Krewplus$ and $\Krewplustilde$ both act on subwords of $\invc^{m}\invc^L$ in the same way, except that the first only considers subwords with product equal to the Coxeter element~$c$, while the second allows products that are in $\NC$.
Secondly, it is not hard to compute their order uniformly for all finite Coxeter groups, see~\cite[Ch.~6]{STW2015}.

\begin{theorem}
\label{thm:order}
  The order of $\Krewplus = (\Krewplus)_m$ and of $\Krewplustilde = (\Krewplustilde)_{m+1}$ is given by
  \[
    \order(\Krewplus) = \order(\Krewplustilde) =
    \begin{cases}
      (m+1)h/2 - 1, & \text{if } \psi \equiv \one \text{ or } \big( W = I_2(k) \text{ for } k \text{ odd and }m=1 \big), \\
      (m+1)h - 2, & \text{otherwise.}
    \end{cases}
  \]
\end{theorem}

We are now in the position to state the ??final main results of this paper.
%CK: Etwas ergaenzt.
In order to be able to state the following theorem, we recall that
the usual $q$-extension of the positive Fu\ss-Catalan numbers is given by
$$\mCatplus(W;q) := \prod_{i=1}^\ell{\frac{[e_i+mh-1]_q}{[e_i+1]_q}},$$
where we use the standard $q$-notation $[\alpha]_q:=(1-q^\alpha)/(1-q) = 1 + q + \dots + q^{\alpha-1}$.

\begin{theorem} \label{thm:CS}
  For a finite Coxeter group~$W$, the triple
  $$\Big(\mNCplus, \Krewplus, \mCatplus(W;q)\Big)$$
  and the triple
  $$\Big(\mNCplus[m+1], \Krewplustilde, \mCatplus[m+1](W;q)\Big)$$
  both exhibit cyclic sieving phenomena as defined in~\cite{RSW2004}.
\end{theorem}

We will prove this theorem using 
%CK: a
a case-by-case analysis in \Cref{sec:typeA,sec:typeB,sec:typeD,sec:typeExc} below.

\section{Positive \texorpdfstring{$m$}{m}-divisible non-crossing partitions in type~$A$}
\label{sec:typeA}

In this section, we study the positive ??Kreweras map in the situation of the symmetric group $\mathcal{S}_n$ corresponding to the Coxeter system
\begin{equation} \label{eq:WSA} 
(W,\reflS) = \big( A_{n-1},\{ s_i = (i,i+1) : 1 \leq i < n\} \big)
\end{equation}
together with the Coxeter element $c = s_1 \cdots s_{n-1}$ being the long cycle $(1,2,\ldots,n)$. In this situation, we have the following simple combinatorial description of positive $m$-divisible non-crossing partitions.
\begin{proposition}
  $(w_0,w_1,\dots,w_m) \in NC^{(m)}(\mathcal{S}_n,c)$ is positive if
  and only if $w_m(n)=n$, or, equivalently, if $(w_0w_1\cdots
  w_{m-1})(n)=1$. 
\end{proposition}

\begin{proof}
According to \Cref{def:NCpos}, the tuple $(w_0,w_1,\dots,w_m)$ is
positive if and only if $cw_m^{-1}=w_0w_1\cdots w_{m-1}$ does not live
in a proper standard parabolic subgroup of $\mathcal{S}_n$, with
respect to our generators $s_i$, $i=1,2,\dots,n-1$, given in
\eqref{eq:WSA}. The
cycles in the disjoint cycle decomposition of $cw_m^{-1}$ define
a non-crossing partition. Thus, $cw_m^{-1}$ will live in a standard
parabolic subgroup of $\mathcal{S}_n$ if and only if 
$(cw_m^{-1})(n)\ne1$. By acting on the left on both sides of
this relation by the inverse of the
Coxeter element $c=(1,2,\dots,n)$, we see that this is equivalent
to $w_m^{-1}(n)\ne n$. We are interested in the contrapositive:
$cw_m^{-1}$ does {\it not\/} live in a standard parabolic subgroup of
$\mathcal{S}_n$ if and only if $w_m(n)=n$.
\end{proof}


Next we recall Armstrong's bijection between $NC^{(m)}(A_{n-1})$ and 
Edelman's $m$-divisible non-crossing partitions of
$\{1,2,\dots,mn\}$, which we denote here by
$\widetilde{NC}{}^{(m)}(A_{n-1})$.
Given an element $(w_0,w_1,\dots,w_m)\in NC^{(m)}(A_{n-1})$,
the bijection, $\Na^m_{A_{n-1}}$ say, 
from \cite[Thm.~4.3.8]{Arm2006} 
works as follows: for $i=1,2,\dots,m$, let $\ta_{m,i}$ be
the transformation which maps a permutation $w\in \mathcal{S}_{n}$ to a
permutation $\ta_{m,i}(w)\in \mathcal{S}_{mn}$ by letting
$$(\ta_{m,i}(w))(mk+i-m)=mw(k)+i-m,\quad k=1,2,\dots,n,$$ 
and 
$(\ta_{m,i}(w))(l)=l$ for all $l\nequiv i$~(mod~$m$).
With the choice of Coxeter element $c=(1,2,\dots,n)$, 
the announced bijection maps 
$(w_0,w_1,\dots,w_m)\in NC^{(m)}(A_{n-1})$ to
\begin{equation} \label{eq:Na}
\Na^m_{A_{n-1}}(w_0,w_1,\dots,w_m)=
(1,2,\dots,mn)\,(\ta_{m,1}(w_1))^{-1}\,(\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\ta_{m,m}(w_m))^{-1},
\end{equation}
where the cycles in the disjoint cycle representation correspond to
the blocks in the non-crossing partition in
$\widetilde{NC}{}^{(m)}(A_{n-1})$.
We refer the reader to \cite[Sec.~4.3.2]{Arm2006} for the details.
For example, let $n=7$, $m=3$, $w_0=(4,5,6)$,
$w_1=(3,6)$,
$w_2=(1,7)$, and
$w_3=(1,2,6)$. Then $(w_0,w_1,w_2,w_3)$ is mapped to
\begin{align} \notag 
\Na^3_{A_{6}}(w_0,w_1,w_2,w_3)&=(1,2,\dots,21)\,(7,16)\,
(2,20)\,(18,6,3)\\
&=(1,2,21)\,(3,19,20)\,(4,5,6)\,(7,17,18)\,(8,9,\dots,16).
\label{eq:NCA}
\end{align} 


\begin{proposition} \label{prop:1}
The image under $\Na^m_{A_{n-1}}$ of the positive $m$-divisible
non-crossing partitions in $NC^{(m)}(A_{n-1})$ are those $m$-divisible
non-crossing partitions in $\widetilde{NC}{}^{(m)}(A_{n-1})$ where $1$ and
$mn$ are in the same block.
\end{proposition}

\begin{proof}
First, let $(w_0,w_1,\dots,w_m)\in NC^{(m)}(A_{n-1})$.
By definition, we know that $w_m(n)=n$. Consequently,
\begin{align*}
\big(\Na^m_{A_{n-1}}(&w_0,w_1,\dots,w_m)\big)(mn)\\
&=
\big((1,2,\dots,mn)\,(\ta_{m,1}(w_1))^{-1}\,(\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\ta_{m,m}(w_m))^{-1}\big)(mn)\\
&=
\big((1,2,\dots,mn)\big)(mn)=1.
\end{align*}
Translated to non-crossing partitions, this means that $mn$ and $1$
belong to the same block, which is what we claimed.

Conversely, let $\pi\in \widetilde{NC}{}^{(m)}(A_{n-1})$ be an
$m$-divisible non-crossing partition in which $1$ and $mn$ are in the
same block. In other words, if $\pi$
is interpreted as a permutation, $\pi(mn)=1$. Let
$(w_0,w_1,\dots,w_m)$ be the element of $NC^{(m)}(A_{n-1})$ such that
$$
\Na^m_{A_{n-1}}(w_0,w_1,\dots,w_m)=\pi.
$$
We must have
$$
\big(\Na^m_{A_{n-1}}(w_0,w_1,\dots,w_m)\big)(mn)=1.
$$
The definition \eqref{eq:Na} of $\Na^m_{A_{n-1}}$
and the fact that $(\ta_{m,i}(w_i))^{-1}$ leaves multiples of $m$
fixed as long as $1\le i\le m-1$ together imply that 
$(\ta_{m,m}(w_m))^{-1}(mn)=mn$, or, equivalently, that
$w_m(n)=n$. By definition, this means that
$(w_0,w_1,\dots,w_m)$ is positive.
\end{proof} 

We let $\widetilde{NC}{}^{(m)}_+(A_{n-1})$ denote the ``positive" elements
of $\widetilde{NC}{}^{(m)}(A_{n-1})$, that is, those which contain $1$ and
$mn$ in the same block. Furthermore, we call this block containing
both $1$ and $mn$ the {\it special block\/} of the partition under
consideration.

Next we have to translate the positive Kreweras map $\Krewplustilde$
into a ``rotation action" on the positive elements of
$NC^{(m)}(A_{n-1})$. In order to do so, we need to explicitly describe 
the decomposition \eqref{eq:LR} in type~$A$.

\begin{lemma} \label{lem:LRA}
Let~$w$ be an element of $\mathcal{S}_n$, and let
$w^{-1}(n)=a$. Then
$$w=w^Lw^R,\quad \quad \text {where $w^R=(a,n)$}.$$
In particular, we have $w^L(n)=n$.
\end{lemma}

\begin{proof}
The last $n-1$ reflections in $\reflR$ given in \eqref{eq:refls}
are $(i,n)$, $i=1,2,\dots,n-1$, in this order. (Recall that we are considering
$\mathcal{S}_n$ which is of type $A_{n-1}$.) Thus, the factorisation
$w=w'\circ(a,n)$ for $a=w^{-1}(n)$ is reduced. This implies that
$w^L=w'$ and $w^R=(a,n)$, as desired.

The final claim of the lemma is obvious.
\end{proof}

Let $m\ge2$. We define the map
$\phi:NC^{(m)}_+(W)\to NC^{(m)}_+(W)$ by
\begin{multline} \label{eq:phi}
(w_0,w_1,\dots,w_m)\\
\mapsto
\big((cw_{m-1}^Rw_mc^{-1})w_0(cw_{m-1}^Rw_mc^{-1})^{-1},
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
\end{multline}
For reasons that will become clear later on,
we shall sometimes call this map {\it pseudo-rotation}.
It is indeed not difficult to see that, if the $(m+1)$-tuple
on the left-hand side is an element of $NC^{(m)}_+(W)$, then so is
the $(m+1)$-tuple on the right-hand side.
Moreover, one can convince oneself (??)
that $\phi^{mn-2}$ acts as the identity.

\begin{remark}
(1) The above definition makes still sense for $m=2$, but it does not for
$m=1$. Whether there is still a sensible way to define $\phi$ even
for $m=1$ remains to be seen.
We shall see that, at least for the types $A_n$, $B_n$, and $D_n$
there are meaningful?? extensions which also cover the case where $m=1$,
see Definitions~??.

\medskip
(2) If $w_0=\one$, then the action $\phi$ reduces to
$$
(\one,w_1,\dots,w_m)
\mapsto
\big(\one,
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
$$
\end{remark}

\begin{proposition} \label{prop:2}
Let $m\ge2$.
Under the bijection $\Na^m_{A_{n-1}}$, the map $\phi$ translates to
the following map $\widetilde\phi$ on $\widetilde{NC}{}^{(m)}_+(A_{n-1})$:
if $mn-1$ is in the same block as $1$ and $mn$ of an element $\pi$
of $\widetilde{NC}{}^{(m)}_+(A_{n-1})$, then $\phi$ rotates all blocks of
$\pi$ by one unit in clockwise direction. On the other hand, if $mn-1$ is
in a different block, say $(b_1,b_2,\dots,mn-1)$ and
$(1,\dots,a,mn)$ are two different blocks, where
$$1<\dots<a<b_1<b_2\dots<mn-1<mn,$$
then the image of $\pi$ under $\widetilde\phi$ is the partition which
contains the blocks $(1,b_2+1,\dots,mn)$ and $(2,\dots,a+1,b_1+1)$
and the blocks which arise from the remaining blocks of $\pi$ by
rotating them by one unit in clockwise direction.
\end{proposition}

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(-3.5,0)}{obj}{45}{2.5}
      {1,,,,,,\hspace{5pt},,,,,,,,,\hspace{5pt},,,,a,,,,,b_1,,,,b_2,,,,,,,\hspace{15pt},,,,,,mn-1\hspace{4pt},,mn\hspace{2pt},}

    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 46pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj7) to[bend right=50] (obj16);
    
    \draw (obj16) to[bend right=50]
          (obj20) to[bend left=10]
          (obj44) to[bend right=50]
          (obj1)  to[bend right=50]
          (obj7);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 38pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj29) to[bend right=50] (obj36);
    
    \draw (obj36) to[bend right=50]
          (obj42) to[bend left=10]
          (obj25) to[bend right=50] (obj29);

    \draw[dotted] (obj2)  to[bend right=50] (obj6);
    \draw[dotted] (obj37) to[bend right=50] (obj41);
    \node at ($($0.92*($(obj4)+(3.5,0)$)$)-(3.5,0)$) {\tiny$X$};
    \node at ($($0.92*($(obj39)+(3.5,0)$)$)-(3.5,0)$) {\tiny$Y$};

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(3.5,0)}{obj}{45}{2.5}
      {1,,2,,,,,,\hspace{5pt},,,,,,,,,\hspace{5pt},,,,a+1,,,,,b_1+1\hspace{8pt},,,,b_2+1\hspace{10pt},,,,,,,\hspace{15pt},,,,,,mn\hspace{2pt},}

    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 46pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj9) to[bend right=50] (obj18);
    
    \draw (obj18) to[bend right=50]
          (obj22) to[bend right=50]
          (obj27) to[bend left=10]
          (obj3)  to[bend right=50]
          (obj9);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 28pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj31) to[bend right=20] (obj38);
    
    \draw (obj38) to[bend right=50]
          (obj44) to[bend right=50]
          (obj1)  to[bend left=10]
          (obj31);

    \draw[dotted] (obj4)  to[bend right=50] (obj8);
    \draw[dotted] (obj39) to[bend right=50] (obj43);
    \node at ($($0.92*($(obj6)-(3.5,0)$)$)+(3.5,0)$) {\tiny$X'$};
    \node at ($($0.92*($(obj41)-(3.5,0)$)$)+(3.5,0)$) {\tiny$Y'$};

    \end{tikzpicture}
  \caption{The action of the pseudo-rotation $\widetilde\phi$}
\label{fig:7}
\end{figure}

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(-3.5,0)}{objleft}{24}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}
     \draw[line width=2.5pt,black] (objleft1) to[bend left] (objleft24);
     \draw[fill=black,fill opacity=0.1] (objleft1) to[bend right=30] (objleft5) to[bend right=30] (objleft6) to[bend right=30] (objleft13) to[bend right=30] (objleft14) to[bend right=30] (objleft24) to[bend right=30] (objleft1);
     \draw[fill=black,fill opacity=0.1] (objleft2) to[bend right=30] (objleft3) to[bend right=30] (objleft4) to[bend left=30] (objleft2);
     \draw[fill=black,fill opacity=0.1] (objleft7) to[bend right=30] (objleft8) to[bend right=30] (objleft12) to[bend left=30] (objleft7);
     \draw[fill=black,fill opacity=0.1] (objleft9) to[bend right=30] (objleft10) to[bend right=30] (objleft11) to[bend left=30] (objleft9);
     \draw[fill=black,fill opacity=0.1] (objleft15) to[bend right=30] (objleft19) to[bend right=30] (objleft23) to[bend left=30] (objleft15);
     \draw[fill=black,fill opacity=0.1] (objleft16) to[bend right=30] (objleft17) to[bend right=30] (objleft18) to[bend left=30] (objleft16);
     \draw[fill=black,fill opacity=0.1] (objleft20) to[bend right=30] (objleft21) to[bend right=30] (objleft22) to[bend left=30] (objleft20);

     \node[inner sep=0pt] at (0,0) {$\mapsto$};

     \polygon{(3.5,0)}{objright}{24}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}
     \draw[line width=2.5pt,black] (objright1) to[bend left] (objright24);
     \draw[fill=black,fill opacity=0.1] (objright2) to[bend right=30] (objright6) to[bend right=30] (objright7) to[bend right=30] (objright14) to[bend right=30] (objright15) to[bend right=30] (objright16) to[bend right=30] (objright2);
     \draw[fill=black,fill opacity=0.1] (objright3) to[bend right=30] (objright4) to[bend right=30] (objright5) to[bend left=30] (objright3);
     \draw[fill=black,fill opacity=0.1] (objright8) to[bend right=30] (objright9) to[bend right=30] (objright13) to[bend left=30] (objright8);
     \draw[fill=black,fill opacity=0.1] (objright10) to[bend right=30] (objright11) to[bend right=30] (objright12) to[bend left=30] (objright10);
     \draw[fill=black,fill opacity=0.1] (objright17) to[bend right=30] (objright18) to[bend right=30] (objright19) to[bend left=30] (objright17);
     \draw[fill=black,fill opacity=0.1] (objright20) to[bend right=30] (objright24) to[bend right=30] (objright1) to[bend left=30] (objright20);
     \draw[fill=black,fill opacity=0.1] (objright21) to[bend right=30] (objright22) to[bend right=30] (objright23) to[bend left=30] (objright21);
    \end{tikzpicture}
  \caption{The action of the pseudo-rotation $\widetilde\phi$}
\label{fig:1}
\end{figure}

\begin{remark}
\Cref{fig:7} provides a schematic illustration of 
the construction in the statement of 
the above proposition, while \Cref{fig:1} provides a concrete
example. In this example, $m=3$, $n=8$, $a=14$,
$b_1=15$, and $b_2=19$.
\end{remark}

\begin{proof}[Proof of \Cref{prop:2}]
The only claim which needs proof is the one on the image of the
blocks $(b_1,b_2,\dots, mn-1)$ and $(1,\dots,a,mn)$ in the case where
$mn-1$ is not in the same block as $1$ and $mn$. By the structure of
$m$-divisible non-crossing partitions, we have 
$$
b_1\equiv b_2-1\equiv a-(m-1)\equiv 0\pmod m.
$$
Let us write 
$b_1=mB_1$,
$b_2=mB_2+1$, and
$a=mA-1$, for suitable integers 
$B_1,B_2,A$. If $(w_0,w_1,\dots,w_m)$ denotes the
element of $NC^{(m)}_+(A_{n-1})$ which corresponds to $\pi$ under the
bijection $\Na^m_{A_{n-1}}$, then we have
\begin{align*}
w_{m-1}(n)&=A,\\
w_{m-1}(B_1)&=n,\\
w_m(B_2)&=B_1,\\
w_m(n)&=n.
\end{align*}
Consequently, we have $w_{m-1}^R=(B_1,n)$ and
\begin{align*}
cw_{m-1}^Rw_mc^{-1}(1)&=B_1+1,\\
cw_{m-1}^Rw_mc^{-1}(B_2+1)&=1,\\
w_{m-1}^L(B_1)&=A,\\
w_{m-1}^L(n)&=n.
\end{align*}
In view of the definition of $\Na^m_{A_{n-1}}$, it follows that
$\phi(w_0,w_1,\dots,w_m)$ maps 
$a+1=mA$ to $mB_1+1=b_1+1$,
$b_1+1=mB_1+1$ to $m\cdot 1-(m-1)+1=2$,
$1=m\cdot 1-(m-1)$ to $m(B_2+1)-(m-1)+1=b_2+1$,
as we claimed.
\end{proof}

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(-4,0)}{obj}{45}{2.5}
      {\hspace{2pt}1,,,,,,a_2,,,,,a_3,,,,,,,,,,,,,,,,,,,a_{k-2}\hspace{8pt},,,,,a_{k-1}\hspace{8pt},,,,,,mn-2\hspace{14pt},,mn-1\hspace{3pt},}

    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 90pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj12) to[bend right=5] (obj31);
    
    \draw (obj31) to[bend right=50]
          (obj36) to[bend right=50]
          (obj44) to[bend right=50]
          (obj1)  to[bend right=50]
          (obj7)  to[bend right=50]
          (obj12);

    \draw[dotted] (obj2)  to[bend right=50] (obj6);
    \draw[dotted] (obj37)  to[bend right=50] (obj41);
    \node at ($($0.92*($(obj4)+(4,0)$)$)-(4,0)$) {\tiny$X$};
    \node at ($($0.92*($(obj39)+(4,0)$)$)-(4,0)$) {\tiny$Y$};

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{45}{2.5}
      {\hspace{2pt}1,,2,,,,,,\hspace{10pt}a_2+1,,,,,\hspace{10pt}a_3+1,,,,,,,,,,,,,,,,,,,a_{k-2}+1\hspace{20pt},,,,,a_{k-1}+1\hspace{20pt},,,,,,mn-1\hspace{3pt},}

    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 90pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj14) to[bend right=5] (obj33);
    
    \draw (obj33) to[bend right=50]
          (obj38) to[bend right=50]
          (obj44) to[bend right=50]
          (obj1)  to[bend right=50]
          (obj9)  to[bend right=50]
          (obj14);

    \draw[dotted] (obj4)  to[bend right=50] (obj8);
    \draw[dotted] (obj39)  to[bend right=50] (obj43);
    \node at ($($0.92*($(obj6)-(4,0)$)$)+(4,0)$) {\tiny$X'$};
    \node at ($($0.92*($(obj41)-(4,0)$)$)+(4,0)$) {\tiny$Y'$};

    \end{tikzpicture}
  \caption{The action of the pseudo-rotation $\widetilde\phi$}
\label{fig:8}
\end{figure}

??general definition of map $\widetilde\phi$, also for $m=1$

\begin{lemma} \label{lem:N-2}
The order of the map $\widetilde\phi$ is $N-2$.
\end{lemma}

\begin{proof}
??
\end{proof}

\subsection{Enumeration}
\label{sec:enumA}

We use the generating function approach from \cite{KratCG}.
We start by recalling the two forms of Lagrange inversion
(cf.\ \cite[Thm.~1.9b]{HenrAA}, \cite[Thm.~5.4.2]{StanBI},
and \cite[Lemma~??]{KratCG}).

\begin{lemma}[\sc Lagrange inversion] \label{lem:TA}
Let $f(z)$ be a formal power series with $f(0)=0$ and $f'(0)\ne0$,
and let $F(z)$ be its compositional inverse. Then, for all integers
$a$ and $b$,
\begin{equation} \label{eq:CA} 
\coef{z^0}z^aF^{b-1}(z)=\coef{z^0}z^bf(z)^{a-1}f'(z)
\end{equation}
and
\begin{equation} \label{eq:CB} 
a\coef{z^0}z^aF^{b}(z)=-b\coef{z^0}z^bf(z)^{a}.
\end{equation}
\end{lemma}

Let $x_i,y_i$, $i=1,2,\dots$, be variables, and $x_0=y_0=1$.
For a positive $m$-divisible non-crossing partition $\pi$ let the weight
$w_{\mathbf y}^{(m)}(\pi)$ be defined by
$$w_{\mathbf y}^{(m)}(\pi)=\prod _{i=1} ^{\infty}
y_i^{\#(\text {special blocks of $\pi$ of size
$mi$})}x_i^{\#(\text {blocks of $\pi$ of size
$mi$})}.$$
For example, the weight $w_{\mathbf y}^{(m)}(\,.\,)$ of the
positive $m$-divisible non-crossing partition on the left of 
\Cref{fig:1} is $y_2x_1^6$, while the weight of the
non-crossing partition on the right of the same figure is
$y_1x_1^5x_2$.

Recall that the special block of a positive $m$-divisible non-crossing
partition $\pi$ is the one which contains both $1$ and $mn$, so that,
clearly, in $w_{\mathbf y}^{(m)}(\pi)$ there appears exactly one variable
$y_i$, and it does so with exponent $1$.
Sometimes, we shall consider the special case where all $y_i$'s are
set equal to $1$. We abbreviate this specialized weight by
$w^{(m)}$, that is,
$$w^{(m)}(\pi)=\prod _{i=1} ^{\infty}x_i^{\#(\text {blocks of $\pi$ of size
$mi$})}.$$
For example, the specialized weight $w^{(m)}(\,.\,)$ of the two
positive $m$-divisible non-crossing partitions in
\Cref{fig:1} is $x_1^6x_2$.


??We write $\vert\pi\vert$ for the {\it size} of $\pi$, that is, 
for the number of elements of the set which is partitioned by $\pi$.

First we quote an auxiliary result from \cite[Lemma~??]{KratCG},
providing a functional equation for the generating function
for {\it all\/}
$m$-divisible non-crossing partitions
with respect to the specialized weight $w^{(m)}$ .

\begin{lemma} \label{lem:TB}
Let $C^{(m)}(z)$ be the generating function $\sum _{\pi}
^{}w^{(m)}(\pi)z^{\vert \pi\vert}$, where the sum is over all
$m$-divisible non-crossing partitions. Then
\begin{equation} \label{eq:CC} 
C^{(m)}(z)=\sum _{i=0} ^{\infty}x_iz^{mi}C^{(m)}(z)^{mi}.
\end{equation}
\end{lemma}


From now on, 
\begin{equation} \label{eq:CD} 
F(z)=\frac {z} {\sum _{i=0} ^{\infty}x_iz^{mi}},
\end{equation}
and $f(z)$ denotes its compositional inverse.

Slightly rewriting \eqref{eq:CC}, we see that it is equivalent to
$$z=\frac {zC^{(m)}(z)} {\sum _{i=0} ^{\infty}x_i(zC^{(m)}(z))^{im}}=
F(zC^{(m)}(z)).$$
Thus, we have $zC^{(m)}(z)=f(z)$.

We quote an easy corollary from \cite[Corollary~??]{KratCG}.

\begin{corollary} \label{cor:TC}
The generating function $\sum _{\pi}
^{}w^{(m)}(\pi)$, where the sum is over all
$m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$, equals
\begin{equation} \label{eq:CE} 
\frac {1} {mn+1}\coef{z^0}zF(z)^{-mn-1},
\end{equation}
where $F(z)$ is defined by \eqref{eq:CD}.
\end{corollary}

Next we turn to the subset of all {\it positive} $m$-divisible non-crossing 
partitions.

\begin{lemma} \label{lem:TBa}
Let $C^{(m)}_+(z)$ be the generating function $\sum _{\pi}
^{}w_{\mathbf y}^{(m)}(\pi)z^{\vert \pi\vert}$, where the sum is over all
positive $m$-divisible non-crossing partitions. Then
\begin{equation} \label{eq:CCa} 
C^{(m)}_+(z)=\sum _{i=1} ^{\infty}y_iz^{mi}C^{(m)}(z)^{mi-1}.
\end{equation}
\end{lemma}

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(0,0)}{obj}{45}{2.5}
      {\hspace{15pt}1\,=\,v_1,,,,,,,,\hspace{5pt}v_2,,,,,,,,\hspace{5pt}v_3,,,,,,,,v_4,,,,,,,,,v_{mi-1}\hspace{15pt},,,,,,,,,,v_{mi}\,{}={}\,mn\hspace{25pt},}
     \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
     \draw (obj1) to[bend right=50] (obj9) to[bend right=50] (obj17) to[bend right=50] (obj25);
     \draw (obj34) to[bend right=50] (obj44) to[bend right=50] (obj1);
     \draw[black, dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 48pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt] (obj25) to[bend right=50] (obj34);
     \draw[dotted] (obj2)  to[bend right=50] (obj8);
     \draw[dotted] (obj10) to[bend right=50] (obj16);
     \draw[dotted] (obj18) to[bend right=50] (obj24);
     \draw[dotted] (obj35) to[bend right=50] (obj43);
     \node at ($0.9*(obj5)$) {\tiny$X_1$};
     \node at ($0.9*(obj13)$) {\tiny$X_2$};
     \node at ($0.9*(obj21)$) {\tiny$X_3$};
     \node at ($0.775*(obj39)$) {\tiny$X_{mi-1}$};
    \end{tikzpicture}
  \caption{The decomposition of a positive $m$-divisible non-crossing partition}
\label{fig:2}
\end{figure}

\begin{proof}
The vertices $1$ and $mn$ 
must be in one of the blocks, say in a block of size $mi$. Then,
if $v_1=1,v_2,\dots,v_{mi}=mn$ are all the vertices in this block in
clockwise order, then the vertices between $v_j$ and $v_{j+1}$,
$j=1,2,\dots,mi-1$ are
involved in a smaller (arbitrary, not necessarily positive) 
$m$-divisible non-crossing partition (see
\Cref{fig:2}). 
Equation~\eqref{eq:CCa} then follows by standard generating function calculus,
the block containing $1$ and $mn$ contributing the term
$y_iz^{mi}$, and the ``small" $m$-divisible non-crossing partitions
between $v_j$ and $v_{j+1}$ each contributing a term
$C^{(m)}(z)$, $j=1,2,\dots,mi-1$.
\end{proof}

The analogue of \Cref{cor:TC} for positive $m$-divisible
non-crossing partitions reads as follows.

\begin{corollary} \label{cor:TCa}
The generating function $\sum _{\pi}
^{}w_{\mathbf y}^{(m)}(\pi)$, where the sum is over all
positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$, equals
\begin{equation} \label{eq:CEa} 
\sum_{i=1}^\infty \frac {mi-1} {mn-1}y_i\coef{z^0}z^{mi-1}F(z)^{-mn+1},
\end{equation}
where $F(z)$ is defined by \eqref{eq:CD}.
\end{corollary}

\begin{proof}
Because of $C^{(m)}(z)=f(z)/z$, we know from \eqref{eq:CCa} 
that the desired generating function 
equals 
\begin{equation} \label{eq:CEb} 
\coef{z^{0}}\sum _{i=1} ^{\infty}y_iz^{-mn+1}f^{mi-1}(z).
\end{equation}
Hence, the claim follows from \eqref{eq:CB} with
$a=mi-1$ and $b=-mn+1$.
\end{proof}

\begin{proposition} \label{prop:TG}
Let $m,n,l$ be positive integers with $l\ge2$. 
Furthermore, let $s'_2,s_3,\dots,s_l$ be
non-negative integers with $s'_2+s_3+\dots+s_l=n-1$. 
The generating function
\begin{equation} \label{eq:DD} 
{\sum }{}^{\displaystyle\prime}
w_{\mathbf y}^{(m)}(\pi_1),
\end{equation}
where the sum is over all multi-chains
$\pi_1\le\pi_2\le\dots\le\pi_{l-1}$ of positive $m$-divisible non-crossing
partitions of $\{1,2,\dots,mn\}$, where the rank of $\pi_i$ is
$s'_2+s_3+\cdots +s_i$, $i=2,3,\dots,l-1$, 
is given by
\begin{multline} \label{eq:DE}
\frac {1} {mn-1}
\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}\\
\times
\sum _{k=0} ^{s_3+\dots+s_l}\!\!
(-1)^{k+s_3+\dots+s_l}
\binom {s_3+\dots+s_l}k 
\sum _{a=1} ^{\infty}
\coef{z^0}(ma-1)y_az^{{ma}+k-1}F(z)^{-{mn}-k+1}.
\end{multline}
If $l=2$, empty sums have to interpreted as $0$, and empty products
have to be interpreted as $1$.
\end{proposition}

\begin{proof}
We prove the assertion by induction on $l$, in the same style as the
proof of Proposition~?? in \cite{KratCG}. If $l=2$, the assertion
is true due to \eqref{eq:CEa} and our conventions for $l=2$. Let us now
assume that the assertion is true for multi-chains consisting of $l-2$
elements and consider the multi-chain
$\pi_1\le\pi_2\le\dots\le\pi_{l-1}$. The induction hypothesis implies that
the generating function 
$${\sum }{}^{\displaystyle\prime}
w^{(m)}_{\mathbf y}(\pi_2),$$
where the sum is over all multi-chains
$\pi_2\le\dots\le\pi_{l-1}$ of positive $m$-divisible non-crossing
partitions of $\{1,2,\dots,mn\}$, where the rank of $\pi_i$ is
$(s'_2+s_3)+s_4+\cdots +s_i$, $i=3,\dots,l-1$, 
is given by
\begin{multline} \label{eq:DF}
\frac {1} {mn-1}\binom {mn-1}{s_4}\cdots\binom {mn-1}{s_{l}}\\
\times
\sum _{k=0} ^{s_4+\dots+s_l}\!\!
(-1)^{k+s_4+\dots+s_l}
\binom {s_4+\dots+s_l}k 
\sum _{a=1} ^{\infty}
\coef{z^0}(ma-1)y_az^{{ma}+k-1}F(z)^{-{mn} -k+1}.
\end{multline}

Next we have to model the relation
$\pi_1\le\pi_2$ by appropriate replacements
of the variables $x_b$ and $y_a$ in the generating function \eqref{eq:DF}.
The partition $\pi_1$ arises from $\pi_2$ by replacing non-special blocks of
$\pi_2$ of size $mb$ by $m$-divisible non-crossing partitions of $mb$
elements, and by replacing the special block, say of size $ma$, 
by a {\it positive} 
$m$-divisible non-crossing partition of
$ma$ elements. Hence, to model this in the
generating function, we will replace in \eqref{eq:DF} the variable $x_b$ by the
coefficient of $u^{mb}$ in $tC^{(m)}(u)$, and we will replace
the variable $y_a$ by the coefficient of $u^{ma}$ in $tC^{(m)}_+(u)$.
(Recall that the series $C^{(m)}(u)$ is
the generating function for {\it all\/} 
$m$-divisible non-crossing partitions, while $C^{(m)}_+(u)$ is the
one for {\it positive} $m$-divisible non-crossing partitions.)
Here, the variable $t$ keeps
track of the number of blocks of $\pi_2$ and, thus, by the equation
$$n-(\text{number of blocks of $\pi_2$})=(\text{rank of $\pi_2$})
=s'_2,$$
respectively equivalently,
$$(\text{number of blocks of $\pi_2$})
=s_3+s_4+\dots+s_l+1,$$
of the rank of $\pi_2$.
From the result we will then extract the coefficient of
$t^{s_3+s_4+\dots+ s_l+1}$ to obtain the generating function \eqref{eq:DD}. 

Copying from \cite[Proof of Prop.~??]{KratCG},
we compute the effect of
substitution of $\coef{u^{mb}}tC^{(m)}(u)$ for $x_{b}$,
$b=1,2,\dots$, in the series $F(z)$:
\begin{align} 
\notag
z\(1+
\sum _{b=1} ^{\infty}z^{mb}\coef{u^{mb}}tC^{(m)}(u)\)^{-1}&=
z\(1+t(C^{(m)}(z)-1)\)^{-1}\\
&=
z\(1+t\(\frac {f(z)} {z}-1\)\)^{-1}.
\label{eq:CHa}
\end{align}
On the other hand, we need to compute the effect of
substitution of $\coef{u^{ma}}tC^{(m)}_+(u)=
\coef{u^{ma}}t\sum _{\ell=1} ^{\infty}y_\ell uf(u)^{m\ell-1}$
(see \eqref{eq:CEa}, respectively \eqref{eq:CEb}) for
$y_a$ in $\sum _{a=1} ^{\infty}(ma-1)y_az^{{ma} -1}$. We obtain
\begin{align} \notag
\sum _{a=1} ^{\infty}(ma-1)z^{{ma} -1}
&\left(\coef{u^{{ma}}}
t\sum _{\ell=1} ^{\infty}y_\ell uf(u)^{m\ell-1}\right)\\
\notag
&=
\sum _{a=1} ^{\infty}\left(\frac {d} {dz}-\frac {1}
     {z}\right)\big(z^{{ma} }\big) \cdot
\left(\coef{u^{{ma}}}
t\sum _{\ell=1} ^{\infty}y_\ell uf(u)^{m\ell-1}\right)\\
\notag
&=
t\left(\frac {d} {dz}\sum _{\ell=1} ^{\infty}
y_\ell zf(z)^{m\ell-1}-
\frac {1} {z}\sum _{\ell=1} ^{\infty} y_\ell zf(z)^{m\ell-1}\right)\\
&=
t
\sum _{\ell=1} ^{\infty}
(m\ell-1)y_\ell zf'(z)f(z)^{m\ell-2}.
\label{eq:DG} 
\end{align}

Using the substitutions \eqref{eq:CHa} and \eqref{eq:DG} in 
\eqref{eq:DF}, subsequent extraction of the
coefficient of $t^{s_3+s_4+\dots+ s_l+1}$ leads to 
\begin{align}
\notag
&\coef{t^{s_3+s_4+\dots+s_l+1}}
\frac {1} {mn-1}
\binom {mn-1}{s_4}\cdots\binom {mn-1}{s_{l}}
\sum _{k=0} ^{s_4+\dots+s_l}\!\!
(-1)^{k+s_4+\dots+s_l}
\binom {s_4+\dots+s_l}k \\
\notag
&\kern2cm
\cdot 
t\sum _{\ell=1} ^{\infty}
\coef{z^0}(m\ell-1)y_\ell zf'(z) f(z)^{{m\ell} -2}z^{-{mn} +1}
\(1+t\(\frac {f(z)} {z}-1\)\)^{{mn} +k-1}\\
\notag
&=\frac {1} {mn-1}
\binom {mn-1}{s_4}\cdots\binom {mn-1}{s_{l}}
\sum _{k=0} ^{s_4+\dots+s_l}\!\!
(-1)^{k+s_4+\dots+s_l}
\binom {s_4+\dots+s_l}k \\
\notag
&\kern.5cm
\cdot 
\sum _{\ell=1} ^{\infty}
\coef{z^0}(m\ell-1)y_\ell f'(z) f(z)^{{m\ell} -2}z^{-{mn} +2}\\
\notag
&\kern5.5cm
\cdot 
\binom {{mn} +k-1}{s_3+s_4+\dots+s_l}
\(\frac {f(z)} {z}-1\)^{s_3+s_4+\dots+s_l}\\
\notag
&=\frac {1} {mn-1}
\binom {mn-1}{s_4}\cdots\binom {mn-1}{s_{l}}%\\
%&\kern.5cm
%\cdot
\sum _{k=0} ^{s_4+\dots+s_l}\!\!
(-1)^{k}
\binom {{mn} +k-1}{s_3+s_4+\dots+s_l}
\binom {s_4+\dots+s_l}k \\
\notag
&\kern2cm
\cdot 
\sum _{\ell=1} ^{\infty}
\sum _{j=0} ^{s_3+s_4+\dots+s_l}(-1)^{s_3+j}
\binom {s_3+s_4+\dots+s_l}j\\
&\kern6cm
\cdot 
\coef{z^0}(m\ell-1)y_\ell f'(z) f(z)^{{m\ell}+j -2}z^{-{mn}-j +2}
\label{eq:DH}
\end{align}
The sums over $k$ and $j$ have
become completely independent, and the sum over $k$ can be
evaluated by means of the Chu--Vandermonde summation formula 
(see e.g.\ \cite[Sec.~5.1, (5.27)]{GrKPAA}). Namely, we have
\begin{align}
\notag
&\sum _{k=0} ^{s_4+\dots+s_l}\!\!
(-1)^k
\binom {mn+k-1}{s_3+s_4+\dots+s_l}
\binom {s_4+\dots+s_l}{k} \\
\notag
&\qquad 
=(-1)^{mn-s_3-s_4-\dots-s_l-1}
\sum _{k=0} ^{s_4+\dots+s_l}\!\!
\binom {-s_3-s_4-\dots-s_l-1}{mn+k-s_3-s_4-\dots-s_l-1}
\binom {s_4+\dots+s_l}{s_4+\dots+s_l-k} \\
\notag
&\qquad 
=(-1)^{mn-s_3-s_4-\dots-s_l-1}
\binom {-s_3-1}{mn-s_3-1}\\
&\qquad 
=(-1)^{s_4+\dots+s_l}
\binom {mn-1}{mn-s_3-1}.
\label{eq:CIa}
\end{align}
If we substitute this in \eqref{eq:DH} and use the Lagrange inversion formula 
\eqref{eq:CA} from \Cref{lem:TA}  with $a= {m\ell}+j-1$ and 
$b=-{mn}
-j+2$, we obtain exactly \eqref{eq:DE}.
\end{proof}

\begin{theorem} \label{thm:countingA}
Let $m,n,l$ be positive integers,
and let $s_1,s_2,\dots,s_l,b_1,b_2,\dots,b_n$ be
non-negative integers with $s_1+s_2+\dots+s_l=n-1$. 
The number of multi-chains 
$\pi_1\le \pi_2\le \dots\le \pi_{l-1}$ in the poset 
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
with the property that $\rk(\pi_i)=s_1+s_2+\cdots +s_i$, 
$i=1,2,\dots,l-1$, and that the
number of blocks of size $mi$ of $\pi_1$ is $b_i$, $i=1,2,\dots,n$,
is given by
\begin{equation} \label{eq:multichains-bi-si} 
\frac {mn-b_1-b_2-\dots-b_n} {(mn-1)(b_1+b_2+\dots+b_n)}
\binom {b_1+b_2+\dots+b_n}{b_1,b_2,\dots,b_n}
\binom {mn-1} {s_2}\cdots
\binom {mn-1} {s_l}
\end{equation}
if $b_1+2b_2+\dots+nb_n=n$ and $s_1+b_1+b_2+\dots+b_n=n$, and\/ $0$ otherwise.
\end{theorem}

\begin{proof}
We use \Cref{prop:TG} with $s'_2=s_1+s_2$. We have
$$n-(\text{number of blocks of $\pi_1$})=(\text{rank of $\pi_1$})
=s_1,$$
or, equivalently,
$$\text{number of blocks of $\pi_1$}
=s_2+s_3+\dots+s_l+1.$$
Hence, we must replace both $x_b$ and $y_b$ by $tx_b$ in 
\eqref{eq:DE} and extract the coefficient of 
$$t^{s_2+s_3+\dots+s_l+1}x_1^{b_1}x_2^{b_2}\cdots x_n^{b_n}.$$
(In the sequel, we write $t^{s_2+s_3+\dots+s_l+1}\mathbf x^{\mathbf b}$ for
short for the above monomial.) 
To begin with, we perform the substitution $y_b\to x_b$ in the
sum over $a$ in \eqref{eq:DE} and simplify:
\begin{align*} %\label{}
\sum _{a=1} ^{\infty}
\coef{z^0}&(ma-1)x_az^{{ma}+k-1}F(z)^{-{mn}-k+1}\\
&=\coef{z^0}z^{k+1}F(z)^{-{mn}-k-1}
\frac {\sum _{a=1} ^{\infty}
(ma-1)x_az^{{ma}}}
{\left(1+\sum _{a=1} ^{\infty}
x_az^{ma}\right)^2}\\
&=\coef{z^0}z^{k+1}F(z)^{-{mn}-k-1}
\left(-F'(z)+z^{-2}F^2(z)\right)\\
&=\coef{z^0}\left(-z^{k+1}F'(z)F(z)^{-{mn}-k-1}
+z^{k-1}F^{-mn-k+1}(z)\right)\\
&=\coef{z^0}\left(z^{k+1}\frac {1} {mn+k}\frac {d} {dz}F(z)^{-{mn}-k}
+z^{k-1}F^{-mn-k+1}(z)\right)\\
&=\coef{z^0}\left(-\frac {k} {mn+k}z^{k}F(z)^{-{mn}-k}
+z^{k-1}F^{-mn-k+1}(z)\right).
\end{align*}
This is now used in \eqref{eq:DE}. Subsequently, as we already said,
we replace $x_b$ by $tx_b$ for all $b$ and extract the coefficient of 
$t^{s_2+s_3+\dots+s_l+1}\mathbf x^{\mathbf b}$. This leads to
\begin{align*}
&\frac {1} {mn-1}
\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}
\sum _{k=0} ^{s_3+\dots+s_l}\!\!
(-1)^{k+s_3+s_4+\dots+s_l}
\binom {s_3+s_4+\dots+s_l}{k} \\
&\kern2cm
\cdot
\coef{t^{s_2+s_3+\dots+s_l+1}\mathbf x^{\mathbf b}z^{mn}}
\Bigg(-\frac {k} {mn+k}\(1+t
\sum _{i=1} ^{\infty}x_iz^{mi}\)^{mn+k}\\
&\kern8.5cm
+\(1+t
\sum _{i=1} ^{\infty}x_iz^{mi}\)^{mn+k-1}\Bigg)\\
&=
\frac {1} {mn-1}
\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}
\sum _{k=0} ^{s_3+\dots+s_l}\!\!
(-1)^{k+s_3+s_4+\dots+s_l}
\binom {s_3+s_4+\dots+s_l}{k} \\
&\kern1cm
\cdot\Bigg(-\frac {k} {mn+k}\binom {mn+k}{s_2+s_3+\dots+s_l+1}
\coef{\mathbf x^{\mathbf b}z^{mn}}
\(\sum _{i=1} ^{\infty}x_iz^{mi}\)^{s_2+s_3+\dots+s_l+1}\\
&\kern3.5cm
+\binom {mn+k-1}{s_2+s_3+\dots+s_l+1}
\coef{\mathbf x^{\mathbf b}z^{mn}}
\(\sum _{i=1} ^{\infty}x_iz^{mi}\)^{s_2+s_3+\dots+s_l+1}
\Bigg)\\
&=\frac {1} {mn-1}
\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}
\sum _{k=0} ^{s_3+\dots+s_l}\!\!
(-1)^{k+s_3+s_4+\dots+s_l}
\binom {s_3+s_4+\dots+s_l}{k} \\
&\kern1cm
\cdot
\frac {-k+(mn+k-s_2-s_3-\dots-s_l-1)} {s_2+s_3+\dots+s_l+1}
\binom {mn+k-1}{s_2+s_3+\dots+s_l}\\
&\kern7cm
\cdot
\coef{\mathbf x^{\mathbf b}z^{mn}}
\(\sum _{i=1} ^{\infty}x_iz^{mi}\)^{s_2+s_3+\dots+s_l+1}.
\end{align*}
The sum over $k$ is completely analogous to the sum over $k$ in the
proof of \Cref{prop:TG} and, hence, can as well be evaluated by
means of the Chu--Vandermonde summation formula. Repeating the
arguments from there, we arrive at the expression
\begin{multline} \label{eq:x^b}
\frac {mn-s_2-s_3-\dots-s_l-1} {(mn-1)(s_2+s_3+\dots+s_l+1)}
\binom {mn-1}{s_2}\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}\\
\times
\coef{\mathbf x^{\mathbf b}z^{mn}}
\(\sum _{i=1} ^{\infty}x_iz^{mi}\)^{s_2+s_3+\dots+s_l+1}.
\end{multline}
Noting that the conditions in the statement of the theorem say that
$$b_1+2b_2+3b_3+\dots+nb_n=n$$
and
$$
s_1+b_1+b_2+\dots+b_n=n,
$$
and, hence,
$$b_1+b_2+\dots+b_n=s_2+s_3+\dots+s_l+1,$$
we arrive at \eqref{eq:multichains-bi-si} without any difficulty.
\end{proof}

The $l=2$ special case of the previous theorem yields an
enumeration formula for positive $m$-divisible non-crossing partitions
with a given block structure.

\begin{corollary} \label{cor:countingA1}
Let $m$ and $n$ be positive integers,
For non-negative integers $b_1,b_2,\dots,b_n$,
the number 
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
which have exactly $b_i$
blocks of size $mi$, $i=1,2,\dots,n$,
is given by
\begin{equation} \label{eq:multichains-bi} 
\frac {1} {mn-1}
\binom {b_1+b_2+\dots+b_n}{b_1,b_2,\dots,b_n}
\binom {mn-1} {b_1+b_2+\dots+b_n}
\end{equation}
if $b_1+2b_2+\dots+nb_n=n$, and\/ $0$ otherwise.
\end{corollary}

On the other hand, if we sum Expression~\eqref{eq:multichains-bi-si} over all
$s_2,s_3,\dots,s_l$ with $s_2+s_3+\dots+s_l=n-s_1-1$, then we obtain
the following generalization.

\begin{corollary} \label{cor:countingA2}
Let $m,n,l$ be positive integers,
For non-negative integers $b_1,b_2,\dots,b_n$,
the number of multi-chains 
$\pi_1\le \pi_2\le \dots\le \pi_{l-1}$ in the poset  
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
for which the number of blocks of size $mi$ of $\pi_1$ is $b_i$,
$i=1,2,\dots,n$,
is given by
\begin{equation} \label{eq:multichains-bia} 
\frac {mn-b_1-b_2-\dots-b_n} {(mn-1)(b_1+b_2+\dots+b_n)}
\binom {b_1+b_2+\dots+b_n}{b_1,b_2,\dots,b_n}
\binom {(l-1)(mn-1)} {b_1+b_2+\dots+b_n-1}
\end{equation}
if $b_1+2b_2+\dots+nb_n=n$, and\/ $0$ otherwise.
\end{corollary}

Summing the result in \Cref{thm:countingA} over all possible block
structures, we arrive at the following result.

\begin{corollary} \label{cor:2}
Let $m,n,l$ be positive integers,
and let $s_1,s_2,\dots,s_l$ be
non-negative integers with $s_1+s_2+\dots+s_l=n-1$. 
The number of multi-chains 
$\pi_1\le \pi_2\le \dots\le \pi_{l-1}$ in the poset 
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
with the property that $\rk(\pi_i)=s_1+s_2+\cdots +s_i$, 
$i=1,2,\dots,l-1$, 
is given by
\begin{equation} \label{eq:multichains-si} 
\frac {mn-s_2-s_3-\dots-s_l-1} {(mn-1)n}
\binom {n}{s_1}
\binom {mn-1} {s_2}\cdots
\binom {mn-1} {s_l}.
\end{equation}
\end{corollary}

\begin{proof}
As indicated above the statement of the corollary, the expression
\eqref{eq:multichains-si} can be obtained from
\eqref{eq:multichains-bi-si} by summing over all possible $b_i$'s,
$i=1,2,\dots,n$, with $b_1+2b_2+\dots+nb_n=n$ and 
$s_1+b_1+b_2+\dots+b_n=n$. A simpler way to arrive at the same result
is to go back to \eqref{eq:x^b}, set all $x_i$'s equal to $1$, to
see that the number which we want to compute is given by
\begin{multline*}
\frac {mn-s_2-s_3-\dots-s_l-1} {(mn-1)(s_2+s_3+\dots+s_l+1)}
\binom {mn-1}{s_2}\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}\\
\times
\coef{z^{mn}}
\(\frac {z^m} {1-z^{mi}}\)^{s_2+s_3+\dots+s_l+1}\\
=\frac {mn-s_2-s_3-\dots-s_l-1} {(mn-1)(s_2+s_3+\dots+s_l+1)}
\binom {mn-1}{s_2}\binom {mn-1}{s_3}\cdots\binom {mn-1}{s_{l}}\\
\times
\binom {n-1}{n-s_2-s_3-\dots-s_l-1},
\end{multline*}
which, because of $s_1+s_2+\dots+s_l=n-1$, is easily seen to be
equivalent to \eqref{eq:multichains-si}.
\end{proof}

Summing the expression in \eqref{eq:multichains-si} over all
possible $s_i$'s with $s_1+s_2+\dots+s_l=n-1$, we obtain the number
of all multi-chains of a given length, and, thus, the {\it zeta
  polynomial} of the poset of positive $m$-divisible non-crossing
partitions of $\{1,2,\dots,mn\}$. As a special case, the {\it total
  number} of positive $m$-divisible non-crossing
partitions of $\{1,2,\dots,mn\}$ falls out.

??M\"obius function

\begin{corollary} \label{cor:3}
Let $m,n,l$ be positive integers,
The number of multi-chains 
$\pi_1\le \pi_2\le \dots\le \pi_{l-1}$ in the poset 
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
is given by
\begin{equation} \label{eq:multichains-l} 
\frac {1+(l-1)(m-1)} {n-1}
\binom {n-1+(l-1)(mn-1)} {n-2}.
\end{equation}
In particular, the total number of elements in this poset equals
\begin{equation} \label{eq:total} 
\frac {1} {n}
\binom {(m+1)n-2} {n-1}.
\end{equation}
\end{corollary}

\begin{proof}
We compute 
\begin{align*}
\sum_{s_1+\dots+s_l=n-1}&
\frac {mn-s_2-s_3-\dots-s_l-1} {(mn-1)n}
\binom {n}{s_1}
\binom {mn-1} {s_2}\cdots
\binom {mn-1} {s_l}\\
&=
\sum_{s_1=0}^{n-1}
\frac {mn-(n-s_1-1)-1} {(mn-1)n}
\binom {n}{s_1}
\binom {(l-1)(mn-1)} {n-s_1-1}\\
&=
\sum_{s_1=0}^{n-1}
\Bigg(\frac {1} {n}
\binom {n}{s_1}
\binom {(l-1)(mn-1)} {n-s_1-1}\\
&\kern2cm
-
\frac {(l-1)(mn-1)} {(mn-1)n}
\binom {n}{s_1}
\binom {(l-1)(mn-1)-1} {n-s_1-2}\Bigg)\\
&=
\frac {1} {n}
\binom {n+(l-1)(mn-1)}{n-1}
-
\frac {l-1} {n}
\binom {(n-1)+(l-1)(mn-1)}{n-2},
\end{align*}
where, to obtain the second line and the last line, 
we used again the Chu--Vandermonde summation formula.
Upon little manipulation, the last expression can be converted into
\eqref{eq:multichains-l}. Formula~\eqref{eq:total} results from
\eqref{eq:multichains-l} by setting $l=2$.
\end{proof}

\subsection{Pseudo-rotationally invariant elements and cyclic sieving}
\label{sec:rotA}

Let $\widetilde R_N$ denote the map sending $\{1,2,\dots,N\}$ into itself
defined by $\widetilde R_N(i)=i+1$~(mod~$N$). In most places, where the
parameter $N$ will be obvious, we shall omit the
index $N$ in $\widetilde R_N$ and shall just write $\widetilde R$ instead.

We call an element of $\widetilde {NC}(N)$ 
{\it $r$-rotationally 
invariant\/} if it is invariant under the action of $\widetilde R^{(N-2)/r}$. 
??Example

We call an element of $\widetilde {NC}_+^{(m)}(n)$ 
{\it $r$-pseudo-rotationally 
invariant\/} if it is invariant under the action of $\widetilde
\phi^{(mn-2)/r}$. 

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(0,0)}{obj}{30}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
     \draw[line width=2.5pt,black] (obj1) to[bend left] (obj30);
     \draw[fill=black,fill opacity=0.1] (obj1) to[bend right=30] (obj2) to[bend right=30] (obj9) to[bend right=30] (obj16) to[bend right=30] (obj23) to[bend right=30] (obj30) to[bend right=30] (obj1);
     \draw[fill=black,fill opacity=0.1] (obj3) to[bend right=30] (obj7) to[bend right=30] (obj8) to[bend left=30] (obj3);
     \draw[fill=black,fill opacity=0.1] (obj4) to[bend right=30] (obj5) to[bend right=30] (obj6) to[bend left=30] (obj4);
     \draw[fill=black,fill opacity=0.1] (obj10) to[bend right=30] (obj14) to[bend right=30] (obj15) to[bend left=30] (obj10);
     \draw[fill=black,fill opacity=0.1] (obj11) to[bend right=30] (obj12) to[bend right=30] (obj13) to[bend left=30] (obj11);
     \draw[fill=black,fill opacity=0.1] (obj17) to[bend right=30] (obj21) to[bend right=30] (obj22) to[bend left=30] (obj17);
     \draw[fill=black,fill opacity=0.1] (obj18) to[bend right=30] (obj19) to[bend right=30] (obj20) to[bend left=30] (obj18);
     \draw[fill=black,fill opacity=0.1] (obj24) to[bend right=30] (obj28) to[bend right=30] (obj29) to[bend left=30] (obj24);
     \draw[fill=black,fill opacity=0.1] (obj25) to[bend right=30] (obj26) to[bend right=30] (obj27) to[bend left=30] (obj25);
    \end{tikzpicture}
  \caption{A $4$-pseudo-rotationally invariant non-crossing partition in
$\widetilde {NC}_+^{(3)}(10)$}
\label{fig:3}
\end{figure}
\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(0,0)}{obj}{30}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
     \draw[line width=2.5pt,black] (obj1) to[bend left] (obj30);
     \draw[fill=black,fill opacity=0.1] (obj3) to[bend right=30] (obj4) to[bend right=30] (obj11) to[bend right=30] (obj18) to[bend right=30] (obj25) to[bend right=30] (obj26) to[bend right=30] (obj3);
     \draw[fill=black,fill opacity=0.1] (obj5) to[bend right=30] (obj9) to[bend right=30] (obj10) to[bend left=30] (obj5);
     \draw[fill=black,fill opacity=0.1] (obj6) to[bend right=30] (obj7) to[bend right=30] (obj8) to[bend left=30] (obj6);
     \draw[fill=black,fill opacity=0.1] (obj12) to[bend right=30] (obj16) to[bend right=30] (obj17) to[bend left=30] (obj12);
     \draw[fill=black,fill opacity=0.1] (obj13) to[bend right=30] (obj14) to[bend right=30] (obj15) to[bend left=30] (obj13);
     \draw[fill=black,fill opacity=0.1] (obj19) to[bend right=30] (obj23) to[bend right=30] (obj24) to[bend left=30] (obj19);
     \draw[fill=black,fill opacity=0.1] (obj20) to[bend right=30] (obj21) to[bend right=30] (obj22) to[bend left=30] (obj20);
     \draw[fill=black,fill opacity=0.1] (obj27) to[bend right=30] (obj28) to[bend right=30] (obj29) to[bend left=30] (obj27);
     \draw[fill=black,fill opacity=0.1] (obj30) to[bend right=30] (obj1) to[bend right=30] (obj2) to[bend left=30] (obj30);
    \end{tikzpicture}
  \caption{Another $4$-pseudo-rotationally invariant non-crossing partition in
$\widetilde {NC}_+^{(3)}(10)$}
\label{fig:4}
\end{figure}
\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(0,0)}{obj}{36}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36}
     \draw[line width=2.5pt,black] (obj1) to[bend left] (obj36);
     \draw[fill=black,fill opacity=0.1] (obj1) to[bend right=30] (obj5) to[bend right=30] (obj12) to[bend right=30] (obj13) to[bend right=30] (obj14) to[bend left=10] (obj36) to[bend right=30] (obj1);
     \draw[fill=black,fill opacity=0.1] (obj18) to[bend right=30] (obj22) to[bend right=30] (obj29) to[bend right=30] (obj30) to[bend right=30] (obj31) to[bend right=30] (obj35) to[bend left=10] (obj18);
     \draw[fill=black,fill opacity=0.1] (obj2) to[bend right=30] (obj3) to[bend right=30] (obj4) to[bend left=30] (obj2);
     \draw[fill=black,fill opacity=0.1] (obj6) to[bend right=30] (obj7) to[bend right=30] (obj11) to[bend left=30] (obj6);
     \draw[fill=black,fill opacity=0.1] (obj8) to[bend right=30] (obj9) to[bend right=30] (obj10) to[bend left=30] (obj8);
     \draw[fill=black,fill opacity=0.1] (obj15) to[bend right=30] (obj16) to[bend right=30] (obj17) to[bend left=30] (obj15);
     \draw[fill=black,fill opacity=0.1] (obj19) to[bend right=30] (obj20) to[bend right=30] (obj21) to[bend left=30] (obj19);
     \draw[fill=black,fill opacity=0.1] (obj23) to[bend right=30] (obj24) to[bend right=30] (obj28) to[bend left=30] (obj23);
     \draw[fill=black,fill opacity=0.1] (obj25) to[bend right=30] (obj26) to[bend right=30] (obj27) to[bend left=30] (obj25);
     \draw[fill=black,fill opacity=0.1] (obj32) to[bend right=30] (obj33) to[bend right=30] (obj34) to[bend left=30] (obj32);
    \end{tikzpicture}
  \caption{A $2$-pseudo-rotationally invariant non-crossing partition in
$\widetilde {NC}_+^{(3)}(12)$}
\label{fig:5}
\end{figure}
\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(0,0)}{obj}{36}{2.5}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36}
     \draw[line width=2.5pt,black] (obj1) to[bend left] (obj36);
     \draw[fill=black,fill opacity=0.1] (obj3) to[bend right=30] (obj7) to[bend right=30] (obj14) to[bend right=30] (obj15) to[bend right=30] (obj16) to[bend right=30] (obj20) to[bend right=10] (obj3);
     \draw[fill=black,fill opacity=0.1] (obj24) to[bend right=30] (obj31) to[bend right=30] (obj32) to[bend right=30] (obj33) to[bend right=30] (obj34) to[bend right=30] (obj2) to[bend left=10] (obj24);
     \draw[fill=black,fill opacity=0.1] (obj4) to[bend right=30] (obj5) to[bend right=30] (obj6) to[bend left=30] (obj4);
     \draw[fill=black,fill opacity=0.1] (obj8) to[bend right=30] (obj9) to[bend right=30] (obj13) to[bend left=30] (obj8);
     \draw[fill=black,fill opacity=0.1] (obj10) to[bend right=30] (obj11) to[bend right=30] (obj12) to[bend left=30] (obj10);
     \draw[fill=black,fill opacity=0.1] (obj17) to[bend right=30] (obj18) to[bend right=30] (obj19) to[bend left=30] (obj17);
     \draw[fill=black,fill opacity=0.1] (obj21) to[bend right=30] (obj22) to[bend right=30] (obj23) to[bend left=30] (obj21);
     \draw[fill=black,fill opacity=0.1] (obj25) to[bend right=30] (obj26) to[bend right=30] (obj30) to[bend left=30] (obj25);
     \draw[fill=black,fill opacity=0.1] (obj27) to[bend right=30] (obj28) to[bend right=30] (obj29) to[bend left=30] (obj27);
     \draw[fill=black,fill opacity=0.1] (obj35) to[bend right=30] (obj36) to[bend right=30] (obj1) to[bend left=30] (obj35);
    \end{tikzpicture}
  \caption{Another $2$-pseudo-rotationally invariant non-crossing partition in
$\widetilde {NC}_+^{(3)}(12)$}
\label{fig:6}
\end{figure}
It can be checked that the positive $3$-divisible non-crossing partitions
in \Cref{fig:3,fig:4} are both $4$-pseudo-rotationally
invariant, that is, they are invariant under the map 
$\widetilde\phi^{(30-2)/4}=\widetilde\phi^7$, while
the positive $3$-divisible non-crossing partitions
in \Cref{fig:5,fig:6} are both $2$-pseudo-rotationally
invariant, that is, they are invariant under the map 
$\widetilde\phi^{(36-2)/2}=\widetilde\phi^{17}$.


Given an element $\pi$ of $\widetilde {NC}^{(m)}(n)$ which is 
$r$-rotationally invariant, a block $B$ of $\pi$ is called
{\it zero block\/} ({\it with respect to $\widetilde R^{(mn-2)/r}$}) 
if $B=\widetilde R^{(mn-2)/r}(B)$.
??Example

For positive $m$-divisible non-crossing partitions which are 
pseudo-rotationally invariant, there is a 
similar concept, although the definition may seem more contrived.
Given an element $\pi$ of $\widetilde {NC}_+^{(m)}(n)$ which is 
$r$-pseudo-rotationally invariant
(with respect to ${\widetilde \phi}^{(mn-2)/r}$), 
a block $B$ of $\pi$ is called
{\it zero block\/} if $\conv(B)\cap \conv(\widetilde R^{(mn-2)/r}(B))\ne\emptyset$,
where $\conv(B)$ means the convex hull of $B$ in the circular representation
of $\pi$, with an analogous meaning of $\conv(\widetilde R^{(mn-2)/r})$.
For example, both non-crossing partitions in \Cref{fig:3,fig:4} 
have zero blocks: in the partition of \Cref{fig:3}
the zero block is $\{1,2,9,16,23,30\}$, while in 
the partition of \Cref{fig:4}
the zero block is $\{3,4,11,18,25,26\}$. On the other hand,
neither of the non-crossing partitions in \Cref{fig:5,fig:6} 
has a zero block.

\begin{lemma} \label{lem:10}
If $r\ge3$,
there cannot be an $r$-pseudo-rotationally invariant positive $m$-divisible
non-crossing partition without zero block.
\end{lemma}

\begin{proof}
??
\end{proof}

%\begin{lemma} \label{lem:11}
%There cannot be an $r$-pseudo-rotationally invariant positive $m$-divisible
%non-crossing partition, in which $mn,1$ are ``isolated" in the
%zero block.
%\end{lemma}

We now describe two constructions of (pseudo-rotationally invariant,
as it will turn out) positive $m$-divisible non-crossing partitions.
In the sequel, $r$ will always be a positive integer dividing $mn-2$
and different from $1$.

\medskip
\noindent
{\sc Construction 1.} 
We start with an (ordinary)
$r$-rotationally invariant 
non-crossing partition of $\{2,3,\dots,mn-1\}$,
all of whose non-zero blocks have size divisible by $m$, $2$ being
contained in the zero block. 
Then we add $1$ and $mn$ to the zero block, thereby creating a
positive $m$-divisible non-crossing partition.
\Cref{fig:3,fig:4} illustrate this construction.
The choice of parameters in these figures is $m=3$ and $n=10$.
In order to obtain the non-crossing partition in \Cref{fig:3},
one starts with the non-crossing partition 
\begin{multline*}
\{\{2,9,16,23\},\{3,7,8\},\{4,5,6\},
\{10,14,15\},\{11,12,13\},\\
\{17,21,22\},\{18,19,20\},
\{24,28,29\},\{25,26,27\}\}
\end{multline*}
of $\{2,3,\dots,29\}$, which is indeed $4$-rotationally invariant
in the above sense. Then one adds $1$ and $3\cdot 10=30$ to the
zero block $\{2,9,16,23\}$ to obtain the partition shown in
\Cref{fig:3}. If one applies $\widetilde\phi$ twice to this
partition, then the result is the partition in \Cref{fig:4}.

\medskip
\noindent
{\sc Construction 2.} 
This construction requires $r$ to equal~2.
We start with an $m$-divisible non-crossing partition $\bar\pi$ of 
$\{mn,1,2,\dots,(mn-2)/2\}$ in which $mn$ and $1$ are in the
same block. Out of these data, we form an $m$-divisible non-crossing
partition $\pi$ of $\{1,2,\dots,mn\}$ in the following way: 
each block of $\bar\pi$ is also a block of $\pi$. Furthermore,
for each
block $B$ of $\bar\pi$ not containing $mn$ and $1$, its ``twin block" 
$\widetilde R^{(mn-2)/2}(B)$ is also a block of $\pi$. Finally, let $B_0$ denote 
the block containing $mn$ and $1$ of $\bar\pi$, say
$B_0=\{mn,1,v_1,v_2,\dots,v_{ma-2}\}$. Then
$\widetilde R^{(mn-2)/2}\big(\{v_1,v_2,\dots,v_{ma-2}\}\big)\cup
\{mn/2,mn-1\}$ is also a block of $\pi$.
\Cref{fig:5,fig:6} illustrate this construction.
The choice of parameters in these figures is $m=3$ and $n=12$.
In order to obtain the non-crossing partition in \Cref{fig:5},
one starts with the non-crossing partition 
$$
\{\{1,5,12,13,14,36\},\{2,3,4\},\{6,7,11\},
\{8,9,10\},\{15,16,17\}\}
$$
of $\{36,1,2,\dots,17\}$. Then the blocks not containing $1$ and 
$3\cdot 12=36$ are rotated, that is, $\widetilde R^{(3\cdot 12-2)/2}=
\widetilde R^{17}$ is applied to them, yielding the blocks
$$
\{19,20,21\},\ \{23,24,28\},\
\{25,26,27\},\ \{32,33,34\}.
$$
All of these become blocks of the partition to be constructed,
see \Cref{fig:5}. Finally, we take the block $B_0$ containing
$1$ and $36$, $B_0=\{1,5,12,13,14,36\}$, remove $1$ and $36$,
apply $\widetilde R^{17}$ to the remaining set,
$$
\widetilde R^{17}(B_0\setminus\{1,36\})=\{22,29,30,31\},
$$
and finally add $3\cdot 12/2=18$ and $3\cdot 12-1=35$ to this
to obtain the last block of the partition to be constructed,
see \Cref{fig:5}.
If one applies $\widetilde\phi$ twice to this
partition, then the result is the partition in \Cref{fig:6}.

\begin{lemma} \label{lem:12}
{\em(1)} If $r\ge3$,
all $r$-pseudo-rotationally invariant positive $m$-divisible non-crossing
  partitions of $\{1,2,\dots,mn\}$ 
are obtained by starting with a non-crossing partition
obtained by Construction~1
and applying the pseudo-rotation $\widetilde\phi$ repeatedly to it.

\smallskip
{\em(2)}
All $2$-pseudo-rotationally invariant positive $m$-divisible non-crossing
  partitions of $\{1,2,\break\dots,mn\}$ 
are obtained by starting with a non-crossing partition
obtained by Construction~{\em1} or Construction~{\em2}
and applying the pseudo-rotation $\widetilde\phi$ repeatedly to it.
\end{lemma}

\begin{proof}
??
\end{proof}


??Problem for chain enumeration: the map $\widetilde \phi$ does not
preserve the partial order on non-crossing partitions.

When we consider $r$-pseudo-rotationally invariant non-crossing
partitions, we need to introduce a ``simplified" weight, which
we denote by $w_{r}^{(m)}$, and which is defined by
$$w^{(m)}_r(\pi)=\prod _{i=1} ^{\infty}
x_i^{\#(\text {non-zero blocks of $\pi$ of size
$mi$})/r}.$$
For example, the weight $w_{4}^{(3)}(\,.\,)$ of both non-crossing partitions
in \Cref{fig:3,fig:4} is $x_1^2$, 
while the weight $w_{2}^{(3)}(\,.\,)$ of both partitions
in \Cref{fig:5,fig:6} is $x_1^4x_2$.

\begin{lemma} \label{lem:TF}
Let $r,m,n,a$ be positive integers with $r\ge2$, $r\mid (mn-2)$, 
and $a\equiv n\ (\text{\rm mod}\ r)$.
The generating function $\sum _{\pi}
^{}w_{r}^{(m)}(\pi)$, where the sum is over all positive
$m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
which are invariant under the $r$-pseudo-rotation
$\widetilde\phi^{(mn-2)/r}$ and have a zero block of
size $ma$ equals
\begin{equation} \label{eq:DA} 
\frac {mn-2} {ma-2}\coef{z^{(mn-2)/r}}f(z)^{(ma-2)/r}=
\coef{z^0}z^{(ma-2)/r}F(z)^{-(mn-2)/r},
\end{equation}
where $F(z)$ is defined by \eqref{eq:CD} and
$f(z)$ is the compositional inverse of $F(z)$, as
before. 
\end{lemma}

\begin{remark}
The above lemma and its proof hold without any change also if the
condition $a\equiv n\ (\text{\rm mod}\ r)$ is relaxed to
$ma\equiv mn\ (\text{\rm mod}\ r)$. However, we shall have no need
of this generalization in this paper.
\end{remark}

\begin{proof}[Proof of \Cref{lem:TF}]
The equality in \eqref{eq:DA} is a direct consequence of the Lagrange
inversion formula \eqref{eq:CB} with $a$ replaced by $(ma-2)/r$ and 
$b=-(mn-2)/r$.
Hence, it suffices to derive one of the two formulae.

\Cref{lem:12}(1) says that --- essentially --- enumerating
positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$ 
which are invariant
under the $r$-pseudo rotation
$\widetilde\phi^{(mn-2)/r}$ is equivalent to enumerating (ordinary)
non-crossing partitions of $mn-2$ elements
with all block sizes except the one of the zero block multiples of
$m$, which are invariant under the
$r$-rotation $\widetilde R^{(mn-2)/r}$.
A careful examination of the proof of Lemma~?? in \cite{KratCG} 
then shows that this enumeration can be achieved by following 
the arguments from there and replacing $mn$ by $mn-2$ and $ma$ by
$ma-2$ everywhere. For the sake of completeness, we outline these
arguments here.

For the following considerations it will be useful to simultaneously
look at Figure~??
which shows a non-crossing partition
of $\{2,3,\dots,29\}$ which is invariant under the $4$-pseudo-rotation
$\widetilde R^{7}$ (in the sense described in Construction~1, that is,
identifying $30$ and $2$) and which has a zero block of size $4=3\cdot 2-2$.
This is the non-crossing partition which underlies the partitions in
\Cref{fig:3,fig:4} through Construction~1.

We want to count non-crossing partitions of $\{2,3,\dots,mn-1\}$
with all block sizes except the one of the zero block multiples of
$m$, the zero block containing the element $2$ and having size $ma-2$,
which remain invariant under an $r$-rotation. The insertion of
$1$ and $mn$ and subsequent action of
$\widetilde \phi$ which is described in \Cref{lem:12}
to obtain {\it all\/} positive $m$-divisible non-crossing partitions
of $\{1,2,\dots,mn\}$ then implies that the result has to
be multiplied by $\frac {(mn-2)/r} {(ma-2)/r}=\frac {mn-2} {ma-2}$.
(In particular, we have to divide by $(ma-2)/r$ in order to avoid
overcounting: any element of the zero block can be moved to the
element $2$ by repeated application of the rotation $\widetilde R$!)

Since the partition is invariant under the $r$-rotation $\widetilde R^{(mn-2)/r}$, 
it suffices to concentrate on the ``fundamental" part of
the partition, which we may choose as the restriction of the partition
to $\mathcal F=\{2,3,\dots,\frac {mn-2} {r}-1\}$.

Now let $s_1=2,s_2,\dots,s_{(ma-2)/r}$ be the elements of the zero block
which are at the same time contained in the fundamental section
$\mathcal F$.
The restriction of our original partition to the vertices in between
$s_i$ and $s_{i+1}$, $i=1,2,\dots,\frac {ma-2} {r}$ (with the convention that
$s_{(ma-2)/r}=s_1+\frac {mn-2} {r}$) is an 
$m$-divisible non-crossing partition on $s_{i+1}-s_i-1$
vertices. That is, the restriction to the fundamental section
$\mathcal F$
consists of $(ma-2)/r$ $m$-divisible non-crossing partitions 
``interleaved" by elements of the zero block.
From \Cref{lem:TB} and the subsequent arguments we know that
the generating function for $m$-divisible non-crossing partitions is
given by $f(z)/z$. Then, by elementary
generating function calculus, we infer that
the generating function $\sum _{\pi}
^{}w_{r}^{(m)}(\pi)$ of the statement in the lemma is equal to
$$\frac {mn-2} {ma-2}\coef{z^{(mn-2)/r}}f(z)^{(ma-2)/r}.$$
This is exactly \eqref{eq:DA}.
\end{proof}

\begin{lemma} \label{lem:TFa}
Let $m$ and $n$ be positive integers with $2\mid mn$.
The generating function $\sum _{\pi}
^{}w_{2}^{(m)}(\pi)$, where the sum is over all positive
$m$-divisible non-crossing partitions of $\{1,2,\break\dots,mn\}$
without zero block
which are invariant under the $2$-pseudo-rotation
$\widetilde\phi^{(mn-2)/2}$ equals
\begin{equation} \label{eq:DAa} 
\sum_{a=1}^\infty 
\frac {mn-2} {2}x_a\coef{z^{(mn-2)/2}}f(z)^{ma-1}=
\sum_{a=1}^\infty 
(ma-1)x_a\coef{z^{-(ma-1)}}F(z)^{-(mn-2)/2},
\end{equation}
where $F(z)$ is defined by \eqref{eq:CD} and
$f(z)$ is the compositional inverse of $F(z)$, as
before. 
\end{lemma}

\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygon{(-4,0)}{obj}{44}{2.5}
      {,,,,v_2,,,,v_3,,,,\hspace{15pt}v_{ma-2},,,,\hspace{13pt}v_{ma-1},,,,,,,,,,,,,,,,,,,,,,,,,,,}

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 18pt on 2pt off 2pt on 2pt off 2pt] (obj9) to[bend right=50] (obj13);
    
    \draw (obj13) to[bend right=50]
          (obj17) to[bend left=50]
          (obj5)  to[bend right=50]
          (obj9);

    \draw[dotted] (obj2)  to[bend right=50] (obj4);
    \node at ($($1.05*($(obj3)+(4,0)$)$)-(4,0)$) {\tiny$X_1$};

    \draw[dotted] (obj6)  to[bend right=50] (obj8);
    \node at ($($1.05*($(obj7)+(4,0)$)$)-(4,0)$) {\tiny$X_2$};

    \draw[dotted] (obj14)  to[bend right=50] (obj16);
    \node at ($($1.15*($(obj15)+(4,0)$)$)-(3.95,0)$) {\tiny$X_{ma-2}$};

    \draw[dotted] (obj18)  to[bend right=50] (obj20);
    \node at ($($1.05*($(obj19)+(4,0)$)$)-(3.75,0)$) {\tiny$X_{ma-1}$};

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{2.5}
      {\hspace{2pt}1\,=\,v_1,,,,v_2,,,,v_3,,,,\hspace{15pt}v_{ma-2},,,,\hspace{13pt}v_{ma-1},,,,\hspace{5pt},,,,\hspace{5pt},,,,\hspace{5pt},,,,\hspace{5pt},,,,\hspace{5pt},,,,mn-1,,mn,}

    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj43);

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 18pt on 2pt off 2pt on 2pt off 2pt] (obj9) to[bend right=50] (obj13);
    
    \draw (obj13) to[bend right=50]
          (obj17) to[bend left=10]
          (obj43) to[bend right=50]
          (obj1)  to[bend right=50]
          (obj5)  to[bend right=50]
          (obj9);

    \draw[dotted] (obj2)  to[bend right=50] (obj4);
    \node at ($($1.05*($(obj3)-(4,0)$)$)+(4,0)$) {\tiny$X_1$};

    \draw[dotted] (obj6)  to[bend right=50] (obj8);
    \node at ($($1.05*($(obj7)-(4,0)$)$)+(4,0)$) {\tiny$X_2$};

    \draw[dotted] (obj14)  to[bend right=50] (obj16);
    \node at ($($1.15*($(obj15)-(4,0)$)$)+(4.05,0)$) {\tiny$X_{ma-2}$};

    \draw[dotted] (obj18)  to[bend right=50] (obj20);
    \node at ($($1.05*($(obj19)-(4,0)$)$)+(4.25,0)$) {\tiny$X_{ma-1}$};

    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 18pt on 2pt off 2pt on 2pt off 2pt] (obj29) to[bend right=50] (obj33);
    
    \draw (obj33) to[bend right=50]
          (obj37) to[bend right=50]
          (obj41) to[bend left=10]
          (obj21) to[bend right=50]
          (obj25) to[bend right=50]
          (obj29);

    \draw[dotted] (obj22)  to[bend right=50] (obj24);
    \node at ($($1.05*($(obj23)-(4,0)$)$)+(4,0)$) {\tiny$X^{'}_{1}$};

    \draw[dotted] (obj26)  to[bend right=50] (obj28);
    \node at ($($1.05*($(obj27)-(4,0)$)$)+(4,0)$) {\tiny$X_{2}^{'}$};

    \draw[dotted] (obj34)  to[bend right=50] (obj36);
    \node at ($($1.15*($(obj35)-(4,0)$)$)+(4.05,0)$) {\tiny$X_{ma-2}^{'}$};

    \draw[dotted] (obj38)  to[bend right=50] (obj40);
    \node at ($($1.05*($(obj39)-(4,0)$)$)+(3.75,0)$) {\tiny$X_{ma-1}^{'}$};

  \end{tikzpicture}
  \caption{The decomposition of a non-crossing partition in 
$\widetilde {NC}_+^{(m)}(n)$ without zero block}
\label{fig:9}
\end{figure}

\begin{proof}
According to \Cref{lem:12}(2), all 
$2$-pseudo-rotationally invariant positive $m$-divisible non-crossing
  partitions of $\{1,2,\dots,mn\}$ 
are obtained by starting with a non-crossing partition
obtained by Construction~2
and applying the pseudo-rotation $\widetilde\phi$ repeatedly to it.
Therefore, what we need to do is to count the number of
non-crossing partitions of $\{1,2,\dots,mn\}$ which can be obtained
by Construction~2, and then multiply the result by $(mn-2)/2$.

If we fix the size of the block containing $mn$ and $1$ in 
Construction~2 to be $ma$, then, by elementary generating function
calculus,  we obtain that
the generating function $\sum _{\pi}
^{}w_{2}^{(m)}(\pi)$ for the non-crossing partitions arising from
Construction~2 with the size of the block containing $mn$ and $1$
being equal to $ma$ is given by (cf.\ \Cref{fig:9})
$$
x_az^{ma}\left(C^{(m)}(z)\right)^{ma-1}
=x_azf^{ma-1}(z),
$$
where $C^{(m)}(z)$ is the generating function for $m$-divisible
non-crossing partitions from \Cref{lem:TB}. Hence, the 
number we want to compute is 
$$
\frac {mn-2} {2}x_a\coef{z^{mn/2}}zf^{ma-1}(z)=
\frac {mn-2} {2}x_a\coef{z^{(mn-2)/2}}f^{ma-1}(z).
$$
This number has to be summed over all possible $a$'s. 
This leads directly to the left-hand side expression in
\eqref{eq:DAa}.
The right-hand side expression is then a consequence of the
Lagrange inversion formula \eqref{eq:CB}.
\end{proof}


\begin{theorem} \label{thm:2}
Let $m,n,r$ be positive integers with $r\ge2$ and $r\mid (mn-2)$.
Furthermore, let $b_1,b_2,\dots,b_n$ be non-negative integers.
The number of positive $m$-divisible non-crossing partitions of 
$\{1,2,\dots,mn\}$ which are invariant under the $r$-pseudo-rotation
$\widetilde \phi^{(mn-2)/r}$, the
number of non-zero 
blocks of size $mi$ being $rb_i$, $i=1,2,\dots,n$,
the zero block having size $ma=mn-mr\sum_{j=1}^njb_j$,
is given by
\begin{equation} \label{eq:multichains-bi-si-pos} 
\binom {b_1+b_2+\dots+b_n}{b_1,b_2,\dots,b_n}
\binom {(mn-2)/r} {b_1+b_2+\dots+b_n}
\end{equation}
if $b_1+2b_2+\dots+nb_n<{n/r}$, 
or if $r=2$ and $b_1+2b_2+\dots+nb_n=n/2$, 
and\/ $0$ otherwise.
\end{theorem}

\begin{proof}
We have to distinguish two cases depending on whether $a$ is non-zero
or not, that is, whether the choice of $b_k$'s implies the existence
of a non-zero block or not.

If $a>0$, that is, $n>r\sum_{j=1}^njb_j$, then,
using the right-hand side expression in \eqref{eq:DA}
and \eqref{eq:CD},
we see that the number which we want to compute is given by
\begin{align*}
\coef{\mathbf x^{\mathbf b}z^0}z^{(ma-2)/r}F(z)^{-(mn-2)/r}
&=
\coef{\mathbf x^{\mathbf b}z^0}z^{(ma-mn)/r}\(1+\sum_{\ell=1}^\infty x_\ell
z^{m\ell}\)^{(mn-2)/r}\\
&=\coef{\mathbf x^{\mathbf b}z^{m(n-a)/r}}
\sum_{k=0}^{(mn-2)/r}\binom {(mn-2)/r}k
\(\sum_{\ell=1}^\infty x_\ell z^{m\ell}\)^k\\
&=
\binom {(mn-2)/r}{b_1+b_2+\dots+b_n}
\binom {b_1+b_2+\dots+b_n}{b_1,b_2,\dots,b_n},
\end{align*}
where the short notation $\mathbf x^{\mathbf b}$ stands again for
$x_1^{b_1}x_2^{b_2}\cdots x_n^{b_n}$.
This is exactly the asserted expression.

On the other hand, if $r=2$ and $n=2\sum_{j=1}^njb_j$, then,
using the right-hand side expression in \eqref{eq:DAa}
and \eqref{eq:CD}, and taking into account that the block
containing $1$ and $mn$ in Construction~2 of size $ma$ contributes
1 to $b_a$,
we see that the number which we want to compute is given by
\begin{align*}
\sum_{a=1}^n
(ma-1)&\coef{x_a^{-1}\mathbf x^{\mathbf b}z^{-(ma-1)}}F(z)^{-(mn-2)/2}\\
&=\sum_{a=1}^n
(ma-1)\coef{x_a^{-1}\mathbf x^{\mathbf b}z^{\frac {mn} {2}-ma}}
\(\sum_{\ell=1}^\infty x_\ell z^{m\ell}\)^{(mn-2)/2}\\
&=\sum_{a=1}^n
(ma-1)\coef{x_a^{-1}\mathbf x^{\mathbf b}z^{\frac {mn} {2}-ma}}
\(1+\sum_{\ell=1}^\infty x_\ell z^{m\ell}\)^{(mn-2)/2}\\
&=\sum_{a=1}^n 
(ma-1)
\binom {(mn-2)/2}{b_1+b_2+\dots+b_n-1}
\binom {b_1+b_2+\dots+b_n-1}
{b_1,\dots,b_{a-1},b_a-1,b_{a+1},\dots,b_n}\\
&=
\binom {(mn-2)/2}{b_1+b_2+\dots+b_n}
\binom {b_1+b_2+\dots+b_n}
{b_1,b_2,\dots,b_n}
\sum_{a=1}^n 
\frac {(ma-1)b_a}
{\frac {mn-2} {2}-b_1-b_2-\dots-b_n}.
\end{align*}
Since $\sum_{a=1}^n {(ma-1)b_a}=\frac {mn}
{2}-b_1-b_2-\dots-b_n$,
this proves the formula in \eqref{eq:multichains-bi-si-pos} also
in this case.
 
According to \Cref{lem:10,lem:12}, in all other cases
there are no $r$-pseudo-rotationally invariant positive $m$-divisible
non-crossing partitions.
\end{proof}

\begin{corollary} \label{cor:4}
Let $m,n,r$ be positive integers with $r\ge2$ and $r\mid (mn-2)$.
Furthermore, let $s$ be a non-negative integer.
The number
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
which are invariant under the $r$-pseudo-rotation $\widetilde\phi^{(mn-2)/r}$,
have $rs$ non-zero blocks and a zero block of size $ma$,
is given by
\begin{equation} \label{eq:multichains-si-pos-a} 
\binom {{(n-a-r)/r}}{s-1}
\binom {(mn-2)/r}{s}
\end{equation}
if $a\equiv n$~{\em(mod~$r$)} and $0$ otherwise,
while the number of these partitions without restricting the size of
the zero block is
\begin{equation} \label{eq:multichains-si-pos} 
\binom {\fl{n/r}}{s}
\binom {(mn-2)/r}{s}.
\end{equation}
\end{corollary}

\begin{proof}
The expression
\eqref{eq:multichains-si-pos-a} could be obtained from
\eqref{eq:multichains-bi-si-pos} by summing over all possible $b_i$'s,
$i=1,2,\dots,n$, with $a=n-r(b_1+2b_2+\dots+nb_n)$ and 
$s+b_1+b_2+\dots+b_n=\fl{n/r}$. It is however simpler to start again with
the right-hand side of \eqref{eq:DA} (if $a>0$), respectively
the right-hand side of \eqref{eq:DAa} (if $a=0$), 
set all $x_j$'s (except $x_0$
which, by definition, equals~$1$) equal to $t$, 
%sum the resulting
%expression over all non-negative integers $a\equiv n$~(mod~$r$),
and finally extract
the coefficient of $t^{s}$. 
Indeed, since in
$r$-pseudo-rotationally invariant $m$-divisible non-crossing partitions
the number of non-zero blocks of size $mi$ must be a multiple of $r$,
implying that the size of the zero block must be $ma=mn-mrB$, for some
positive integer $B$, we have
% it suffices indeed to sum the expression
%\eqref{eq:DA} over 
$a\equiv n$~(mod~$r$).
%in order to cover all possible sizes of zero blocks.
Hence, if $a>0$, from \eqref{eq:DA} we obtain
\begin{align*}
\coef{t^{s}z^0}
&z^{(ma-2)/r}F(z)^{-(mn-2)/r}\Big\vert_{x_1=x_2=\dots=t}\\
&=
\coef{t^{s}z^0}
z^{(ma-mn)/r}\(1+t\sum_{\ell=1}^\infty 
z^{m\ell}\)^{(mn-2)/r}\\
&=\coef{t^{s}z^{(mn-ma)/r}}
\sum_{k=0}^{(mn-2)/r}\binom {(mn-2)/r}k
t^k\(\frac {z^m} {1-z^m}\)^k\\
&=\coef{z^{m\frac {n-a} {r}-ms}}
\binom {(mn-2)/r}{s}
\(\frac {1} {1-z^m}\)^{s}\\
&=
\binom {(mn-2)/r}{s}
\binom {{(n-a-r)/r}}{s-1},
\end{align*}
%\begin{align*}
%\coef{t^{s}z^0}\sum_{a\equiv n~\text{(mod $r$)}}
%&z^{(ma-2)/r}F(z)^{-(mn-2)/r}\Big\vert_{x_1=x_2=\dots=t}\\
%&=
%\coef{t^{s}z^0}\sum_{a\equiv n~\text{(mod $r$)}}
%z^{(ma-mn)/r}\(1+t\sum_{\ell=1}^\infty 
%z^{m\ell}\)^{(mn-2)/r}\\
%&=\coef{t^{s}z^{0}}
%\frac {z^{-m\fl{\frac nr}}} {1-z^m}
%\sum_{k=0}^{(mn-2)/r}\binom {(mn-2)/r}k
%t^k\(\frac {z^m} {1-z^m}\)^k\\
%&=\coef{z^{ms}}
%\binom {(mn-2)/r}{\fl{n/r}-s}
%\(\frac {1} {1-z^m}\)^{\fl{n/r}-s+1}\\
%&=
%\binom {(mn-2)/r}{\fl{n/r}-s}
%\binom {\fl{n/r}}{s},
%\end{align*}
which is exactly \eqref{eq:multichains-si-pos-a}.
By summing this expression over all $a\equiv n$~(mod~$r$),
we obtain \eqref{eq:multichains-si-pos}.
\end{proof}

\begin{corollary} \label{cor:5}
Let $m,n,a,r$ be positive integers with $r\ge2$ and $r\mid (mn-2)$.
The number
of positive $m$-divisible non-crossing partitions of $\{1,2,\dots,mn\}$
which are invariant under the $r$-pseudo-rotation
$\widetilde\phi^{(mn-2)/r}$ and have a zero block of size $ma$
is given by
\begin{equation} \label{eq:multichains-pos-a} 
\binom {{((m+1)n-a-r-2)/r}}{{(n-a)/r}}
\end{equation}
if $a\equiv n$~{\em(mod~$r$)} and $0$ otherwise,
while the number of these partitions without restricting the size of
the zero block is
\begin{equation} \label{eq:multichains-pos} 
\binom {\fl{((m+1)n-2)/r}}{\fl{n/r}}.
\end{equation}
\end{corollary}

\begin{proof}
One sums Expressions~\eqref{eq:multichains-si-pos-a} and
\eqref{eq:multichains-si-pos} over all $s$.
The resulting sums can be evaluated by means of the Chu--Vandermonde
summation formula.
\end{proof}

\subsubsection*{Cyclic sieving}

We use the standard $q$-notations $[\alpha]_q:=(1-q^\alpha)/(1-q)$,
$[n]_q!:=[n]_q\, [n-1]_q\cdots[1]_q$, where $[0]_q!:=1$,
the $q$-binomial coefficient
$$
\begin{bmatrix} M\\m\end{bmatrix}_q:=\begin{cases}
\frac {[M]_q\, [M-1]_q\cdots [M-m+1]_q} {[m]_q!},&\text{if $k\ge1$},\\
1,&\text{if $k=0$,}
\end{cases}
$$
and the $q$-multinomial coefficient
$$
\begin{bmatrix} {M}\\{m_1,m_2,\dots,m_r}\end{bmatrix}_q
:=\begin{cases}
\frac {[M]_q\,[M-1]_q\cdots [M-m_1-m_2-\dots-m_r+1]_q} {[m_1]_q!\,[m_2]_q!\cdots
[m_r]_q!},&\text {if $m_1+\dots+m_r\ge1$,}\\
1,&\text {if $m_1+\dots+m_r=0$.}
\end{cases}
$$

We start with a strengthening of a non-negativity result of Andrews
\cite[Thm.~2]{AndrCB}. 
The proof which we give is a slightly economized form of the original
proof of Andrews. 
We shall frequently refer
to its line of argument later on.

\begin{lemma} \label{lem:Andrews}
Let $a$ and $b$ be positive integers.
Then $\frac {[\gcd(a,b)]_q} {[a+b]_q}\left[\smallmatrix
a+b\\a\endsmallmatrix\right]_q$ is a polynomial in $q$ with non-negative
integer coefficients.
\end{lemma}
\begin{proof}
We first show that the expression in
$\frac {[\gcd(a,b)]_q} {[a+b]_q}\left[\smallmatrix
a+b\\a\endsmallmatrix\right]_q$ is a polynomial in $q$, and then derive the
non-negativity of the coefficients from the fact that the $q$-binomial
coefficient 
$\left[\begin{smallmatrix} a+b\\a\end{smallmatrix}\right]_q$ is a 
reciprocal\footnote{A polynomial
$P(q)$ in $q$ of degree $d$ is called {\it reciprocal\/} if
$P(q)=q^dP(1/q)$.} and unimodal\footnote{\label{foot:2}%
A polynomial $P(q)=
\sum _{i=0} ^{d}p_iq^i$ is called {\it unimodal\/} if there is an
integer $r$ with $0\le r\le d$ and $0\le p_0\le\dots\le p_r\ge\dots\ge
p_d\ge0$. It is well-known that $q$-binomial coefficients are 
unimodal, see \cite[Ex.~7.75.d]{StanBI}.} polynomial in $q$. 
In order to show polynomiality, one recalls the well-known fact that
$$
q^n-1=\prod _{d\mid n} ^{}C_d(q),
$$
where $C_d(q)$ denotes the $d$\th\ cyclotomic polynomial in $q$.
Consequently, 
$$
\frac {[\gcd(a,b)]_q} {[a+b]_q}\begin{bmatrix} a+b\\a\end{bmatrix}_q=
\prod _{d=2} ^{a+b-1}C_d(q)^{e_d},
$$
with
\begin{equation} \label{eq:SH} 
e_d=\chi(d\mid \gcd(a,b))
+\fl{\frac {a+b-1} {d}}
-\fl{\frac {a} {d}}
-\fl{\frac {b} {d}},
\end{equation}
where $\chi(\mathcal S)=1$ if $\mathcal S$ is
true and $\chi(\mathcal S)=0$ otherwise.
Let us write 
$A=\{a/d\}$ and
$B=\{b/d\}$,
where $\{\al\}:=\al-\fl{\al}$ denotes the fractional
part of $\al$.
Using this notation, Equation~\eqref{eq:SH} becomes
$$e_d=\chi(d\mid \gcd(a,b))+\fl{A+B-\frac {1} {d}}.$$ 
This is clearly non-negative, unless 
$A=B=0$. However, in that case we have $d\mid a$ and $d\mid b$,
that is, $d\mid\gcd(a,b)$, so that $e_d$ is non-negative
also in this case.

Since we know that 
$\left[\begin{smallmatrix} a+b\\a\end{smallmatrix}\right]_q$ is a 
reciprocal and unimodal polynomial in $q$ of degree $ab$
(see Footnote~\ref{foot:2}), 
the first $\fl{ab/2}$ coefficients of
$(1-q^{\gcd(a,b)})\left[\begin{smallmatrix} a+b\\a\end{smallmatrix}\right]_q$
are non-negative, and therefore the same must be true for
$$
\frac {1-q^{\gcd(a,b)}} {1-q^{a+b}}
\begin{bmatrix} a+b\\a\end{bmatrix}_q
=\frac {[\gcd(a,b)]_q} {[a+b]_q}\begin{bmatrix} a+b\\a\end{bmatrix}_q.
$$
However, we already know that the last expression 
is actually a {\it polynomial} in $q$, of degree
$ab-(a+b)+\gcd(a,b)$, and it is as well a
reciprocal polynomial. Hence, also the remaining coefficients are
non-negative, which is what we wanted to prove.
\end{proof}


\begin{lemma} \label{lem:1}
Let $m,n$ be positive integers and $s_1,s_2,\dots, s_l,b_1,b_2,\dots,b_n$ 
non-neg\-ative integers such that
\begin{align}
\label{eq:SA}
s_1+s_2+\dots+s_l&=n-1,
\\
\label{eq:SB}
b_1+2b_2+\dots+nb_n&=n,
\\
\label{eq:SC}
s_1+b_1+b_2+\dots+b_n&=n.
\end{align}
Then the expressions
\begin{gather}
\frac {1} {[n]_q}\begin{bmatrix} mn-2\\n-1\end{bmatrix}_q,
\label{eq:SD}
\\
\frac {[mn-s_2-s_3-\dots-s_l-1]_q} {[mn-1]_q\,[n]_q}
\begin{bmatrix} {n}\\{s_1}\end{bmatrix}_q
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q,
\label{eq:SE}
\\
\frac {[mn-b_1-b_2-\dots-b_n]_q} {[mn-1)]_q\,[b_1+b_2+\dots+b_n]_q}
\begin{bmatrix} {b_1+b_2+\dots+b_n}\\{b_1,b_2,\dots,b_n}\end{bmatrix}_q
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q
\label{eq:SF}
\end{gather}
are polynomials in $q$ with non-negative integer coefficients.
\end{lemma}

\begin{proof}
Expression~\eqref{eq:SD} is the special case of \eqref{eq:SE} where $l=2$, $s_1=0$ and
$s_2=n-1$, and consequently does not need to be treated separately.


\medskip
In order to establish polynomiality of Expression~\eqref{eq:SE},
we rewrite the expression \eqref{eq:SE} in the form
{\refstepcounter{equation}\label{eq:SEa}}
\alphaeqn
\begin{gather} \label{eq:SEa.1}
\frac {[mn-s_2-s_3\dots-s_l-1]_q} 
{[\gcd(mn-1,s_2,s_3,\dots,s_l)]_q\,[\gcd(n,s_1)]_q}
\kern4cm
\\
\label{eq:SEa.2}
\times
\frac {[\gcd(n,s_1)]_q} {[n]_q}
\begin{bmatrix} {n}\\{s_1}\end{bmatrix}_q\\
\label{eq:SEa.3}
\kern4cm
\times
\frac {[\gcd(mn-1,s_2,s_3,\dots,s_l)]_q} {[mn-1]_q}
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q. 
\end{gather}
\reseteqn
We note that, because of \eqref{eq:SA},
both  $\gcd(n,s_1)$ and $\gcd(mn-1,s_2,s_3,\dots,s_l)$ 
divide $mn-s_2-s_3-\dots-s_l-1$. Moreover, we claim that the two numbers
are coprime. For, if there is a $d$ dividing both 
$\gcd(n,s_1)$ and $\gcd(mn-1,s_2,s_3,\dots,s_l)$, then from \eqref{eq:SA}
we infer that $d\mid (n-1)$, which, in combination with $d\mid n$, means
that actually $d=1$.
It is then not difficult to see
that the quotient in \eqref{eq:SEa.1} is a polynomial in $q$
(see e.g.\ \cite[Cor.~6]{KrMuAD}). 

Clearly, \Cref{lem:Andrews} says that \eqref{eq:SEa.2}
is a polynomial. In order to see that \eqref{eq:SEa.3} is
also a polynomial, we write
$$
\frac {[\gcd(mn-1,s_2,s_3,\dots,s_l)]_q} {[mn-1]_q}
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q=
\prod _{d=2} ^{mn-1}C_d(q)^{e_d},
$$
with
\begin{multline*} %\label{eq:SHa} 
e_d=\chi(d\mid \gcd(mn-1,s_2,s_3,\dots,s_l))
+\fl{\frac {mn-2} {d}}\\
+(l-2)\fl{\frac {mn-1} {d}}
-\sum_{j=2}^l \left(\fl{\frac {s_j} {d}}+\fl{\frac {mn-s_j-1} {d}}
\right).
\end{multline*}
Letting
$M=\{(mn-1)/d\}$,
$S_2=\{s_2/d\}$, etc., this becomes
\begin{equation} \label{eq:SHa} 
e_d=\chi(d\mid \gcd(mn-1,s_2,s_3,\dots,s_l))
+\fl{M-\frac {1} {d}}
-\sum_{j=2}^l \fl{M-S_j}.
\end{equation}
This is non-negative, unless $M=0$ (in which case
$\fl{M-\frac {1} {d}}=-1$) and $S_2=\dots=S_l=0$. 
But, in that case, we have $d\mid (mn-1)$ and $d\mid s_j$ for all $j$
with $2\le j\le l$, so that $\chi(d\mid \gcd(mn-1,s_2,s_3,\dots,s_l))=1$.
Therefore, the right-hand side of \eqref{eq:SHa} is even non-negative in
this case.

\medskip
Finally, to see that \eqref{eq:SE} is not only a polynomial, but also
has non-negative coefficients, we proceed as in the proof of
\Cref{lem:Andrews}. Since
products of reciprocal and unimodal polynomials are again reciprocal
and unimodal (see \cite[Thm.~3.9]{AndrAF}), the polynomials
$$
\begin{bmatrix} {n}\\{s_1}\end{bmatrix}_q
\quad \text{and}\quad 
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q
$$
are reciprocal and unimodal. The first one is of degree $D_1:=s_1(n-s_1)$,
while the second is of degree $D_2:=\sum_{j=2}^l s_j(mn-s_j-1)$. 
As a consequence, the first $\fl{D_1/2}$ coefficients of 
$(1-q)\left[\begin{smallmatrix} {n}\\{s_1}\end{smallmatrix}\right]_q$
and the first $\fl{D_2}$ coefficients of
$$
(1-q^{mn-s_2-s_3-\dots-s_l-1})
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q
$$
are non-negative. But then the first $\fl{D_1}+\fl{D_2}$
coefficients of the series
$$
\frac {(1-q)(1-q^{mn-s_2-s_3-\dots-s_l-1})} {(1-q^{mn-1})(1-q^n)}
\begin{bmatrix} {n}\\{s_1}\end{bmatrix}_q
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q,
$$
are also non-negative. 
However, we know that the last expression (which agrees with \eqref{eq:SE}) 
is actually a {\it polynomial} in $q$, of degree
$D_1+D_2-(n-1)-(s_2+s_3+\dots+s_l)$, and it is as well a
reciprocal polynomial. Hence, also the remaining coefficients are
non-negative, which is what we wanted to prove.

\medskip
The claims about \eqref{eq:SF} can be established in a completely analogous
fashion. To prove polynomiality, we rewrite \eqref{eq:SF} in the form
{\refstepcounter{equation}\label{eq:SFa}}
\alphaeqn
\begin{gather} \label{eq:SFa.1} 
\frac {[mn-b_1-b_2-\dots-b_n]_q} 
{[\gcd(b_1,b_2,\dots,b_n)]_q\,[\gcd(mn-1,s_2,s_3,\dots,s_l)]_q}
\kern4cm\\
\label{eq:SFa.2} 
\times
\frac {[\gcd(b_1,b_2,\dots,b_n)]_q} {[b_1+b_2+\dots+b_n]_q}
\begin{bmatrix}
  {b_1+b_2+\dots+b_n}\\{b_1,b_2,\dots,b_n}\end{bmatrix}_q\\
\label{eq:SFa.3} 
\kern4cm
\times
\frac {[\gcd(mn-1,s_2,s_3,\dots,s_l)]_q} {[mn-1)]_q}
\begin{bmatrix} {mn-1}\\ {s_2}\end{bmatrix}_q\cdots
\begin{bmatrix} {mn-1}\\ {s_l}\end{bmatrix}_q
\end{gather}
\reseteqn
It follows from \eqref{eq:SB} that $\gcd(b_1,b_2,\dots,b_n)$ divides
$(mn-b_1-b_2-\dots-b_n)$. On the other hand, a combination of
\eqref{eq:SA} and \eqref{eq:SC} leads to the equation
\begin{equation*} %\label{eq:bi-si}
b_1+b_2+\dots+b_n-1=s_2+s_3+\dots+s_l. 
\end{equation*}
Hence, also $\gcd(mn-1,s_2,s_3,\dots,s_l)$ divides
$(mn-b_1-b_2-\dots-b_n)$.
Moreover, if there would be a $d$ dividing both
$\gcd(b_1,b_2,\dots,b_n)$ and
$\gcd(mn-1,s_2,s_3,\dots,s_l)$, then \eqref{eq:SB} would imply that $d\mid
n$, so that, if this is combined with $d\mid (mn-1)$, we see that we
must have $d=1$. As earlier, as a consequence, the fraction
\eqref{eq:SFa.1} is a polynomial in $q$. That \eqref{eq:SFa.2} is a
polynomial in $q$ can be seen by proceeding along the lines of the
proof of \Cref{lem:Andrews}. We leave it to the reader to work
out the details. The expression in \eqref{eq:SFa.3} 
is identical with the one in \eqref{eq:SEa.3}, of which we already
know that it is a polynomial in $q$. This establishes the
polynomiality of \eqref{eq:SF}. Non-negativity of the coefficients
follows now as before of those of \eqref{eq:SE}. We omit 
the details for the sake of brevity.
\end{proof}

\begin{lemma} \label{lem:2}
Let $m,n,s_1,s_2,\dots, s_l,b_1,b_2,\dots,b_n$ be integers satsifying
the hypotheses in \Cref{lem:1}. 
Furthermore, let $p$ be a positive integer such that $p\mid mn-2$ and 
$p<mn-2$. Writing $r=(mn-2)/p$ and $\om=e^{2\pi i/r}$, we have
\begin{align}
\frac {1} {[n]_q}\bmatrix (m+1)n-2\\n-1\endbmatrix
\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}
&=\binom {\fl{((m+1)n-2)/r}} {\fl{n/r}},
\label{eq:SI}
\\
\frac {1} {[n]_q}\bmatrix mn-2\\n-1\endbmatrix
\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}
&=\begin{cases}
\binom {(mn-2)/r} {\fl{n/r}},&\text{if $n\equiv 0,1$ \rm (mod $r$)},\\
0,&\text{otherwise,}
\end{cases}
\label{eq:SJ}
\\
\notag
\frac {[mn-s_2-s_3-\dots-s_l-1]_q} {[mn-1]_q\,[n]_q}
&\bmatrix {n}\\{s_1}\endbmatrix_q
\bmatrix {mn-1}\\ {s_2}\endbmatrix_q\cdots
\bmatrix {mn-1}\\ {s_l}\endbmatrix_q
\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}
&\\
&\hskip-5cm
=\begin{cases}
\frac {(1-\om^{\overline s_1-\overline n+2})
(1-\om^{\overline n-1})(1-\om^{\overline n-2})\cdots
(1-\om^{\overline n-\overline s_1+1})} 
{(1-\om^{\overline s_1})(1-\om^{\overline s_1-1})\cdots(1-\om)}\\
\quad \times
\binom {\fl{n/r}} {\fl{s_1/r}}
\binom {(mn-2)/r} {\fl{s_2/r}}\cdots
\binom {(mn-2)/r} {\fl{s_l/r}},
&\text{if $s_j\equiv 0,1$ \rm (mod $r$) for $j\ge2$},\\
&\{s_1/r\}\le\{n/r\},\\
& \text{not }n\equiv s_1\equiv0\text{
    \em{(mod $r$)},}\\
&\text{and\/ $r\nmid(n-s_1-2)$},\\
\frac {mn-n+s_1} {n}\binom {n/2} {{s_1/2}}
\binom {(mn-2)/2} {\fl{s_2/2}}\cdots
\binom {(mn-2)/2} {\fl{s_l/2}},
&\text{if $r=2$, $n\equiv s_1\equiv0$ \rm (mod $2$),}\\
0,&\text{otherwise,}
\end{cases}
\notag
\\
\label{eq:SK}
\\
\notag
&\hskip-4.5cm
\frac {[mn-b_1-b_2-\dots-b_n]_q} {[mn-1]_q\,[b_1+b_2+\dots+b_n]_q}
\bmatrix {b_1+b_2+\dots+b_n}\\{b_1,b_2,\dots,b_n}\endbmatrix_q
\bmatrix {mn-1}\\ {s_2}\endbmatrix_q\cdots
\bmatrix {mn-1}\\ {s_l}\endbmatrix_q\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}\\
&\hskip-4cm=\begin{cases}
\frac {(1-\om^{2-\sum_{k=1}^n\overline b_k})
(1-\om^{\sum_{k=1}^n\overline b_k-1})\cdots (1-\om)} 
{
\prod _{k=1} ^{n}\big((1-\om ^{\overline b_k})\cdots (1-\om)\big)}
\\
\quad \times\binom {\fl{(b_1+b_2+\dots+b_n)/r}}
{\fl{b_1/r},\fl{b_2/r},\dots,\fl{b_n/r}}
\binom {(mn-2)/r} {\fl{s_2/r}}\cdots
\binom {(mn-2)/r} {\fl{s_l/r}},
&\text{if $s_j\equiv0,1$ \rm (mod $r$), $2\le j\le l$},\\
&\hskip-3cm
\text{$1\ge\sum_{k=1}^n \{b_k/r\}\ge\frac {1} {r}$, and\/
$r\nmid \left(2-\sum_{k=1}^n b_k\right)$},\\
\frac {mn-b_1-\dots-b_n} {b_1+b_2+\dots+b_n}
\binom {{(b_1+b_2+\dots+b_n)/r}}
{{b_1/r},{b_2/r},\dots,{b_n/r}}
\binom {(mn-2)/r} {\fl{s_2/r}}\cdots
\binom {(mn-2)/r} {\fl{s_l/r}},\hskip-4cm\\
&\hskip-3cm
\text{if $r=2$, $n\equiv b_1\equiv\dots\equiv b_n\equiv0$ \rm (mod $2$),}\\
0,&\text{otherwise.}
\end{cases}
\notag
\\
\label{eq:SL}
\end{align}
\end{lemma}


\begin{proof} 
We have
$$
\frac {1} {[n]_q}\bmatrix (m+1)n-2\\n-1\endbmatrix
=\frac {(1-q^{(m+1)n-2})(1-q^{(m+1)n-3})\cdots(1-q^{mn})} 
{(1-q^{n})(1-q^{n-1})\cdots(1-q^{2})}.
$$
In order to perform the limit $q\to e^{2\pi ip/(mn-2)}$, we use 
the simple fact 
\begin{equation} \label{eq:SM}
\lim_{q\to e^{2\pi ip/(mn-2)}} \frac {1-q^\al} {1-q^\be}=
\begin{cases}
\frac \alpha \beta,&\text{if }\alpha\equiv\beta\equiv0\pmod r,\\
1,&\text{otherwise},
\end{cases}	
\end{equation}
where $\alpha,\beta$ are
non-negative integers such that 
$\alpha\equiv\beta\pmod r$. Since $r\mid mn-2$, the claimed limit
in \eqref{eq:SI} is obtained straightforwardly.

\medskip
In order to see \eqref{eq:SJ}, we write
\begin{equation} \label{eq:SN}
\frac {1} {[n]_q}\bmatrix mn-2\\n-1\endbmatrix
=\frac {(1-q^{mn-2})(1-q^{mn-3})\cdots(1-q^{mn-n})} 
{(1-q^{n})(1-q^{n-1})\cdots(1-q^{2})}.
\end{equation}
Here, we have to first check in which cases the number of factors
in the numerator
which vanish when $q\to e^{2\pi ip/(mn-2)}$ does not exceed
the number of those
in the denominator, because otherwise the limit of the left-hand side 
of \eqref{eq:SN} as $q\to e^{2\pi ip/(mn-2)}$ is zero. 
Now, the difference of the numbers of vanishing factors
in the numerator and the denominator is
$$
\fl{\frac {mn-2} {r}}-\fl{\frac {mn-n-1} {r}}-\fl{\frac {n} {r}}=
-\fl{\frac {-n+1} {r}}-\fl{\frac {n} {r}}.
$$
Writing $N=\{n/r\}$ for the fractional part of $n/r$, this reduces to
$
-\fl{-N+\frac {1} {r}},
$
which can only be zero if $N=0$ or $N=1/r$, or, equivalently,
$n\equiv0,1$~(mod~$r$). In these two cases, the assertion
\eqref{eq:SJ}
follows immediately from \eqref{eq:SM}.

\medskip
Next we consider \eqref{eq:SK}. We rewrite the expression on the left-hand
side in an analogous fashion as product/quotient of factors of the
form $1-q^j$. Then, again, we count how many of these factors 
vanish when $q\to e^{2\pi ip/(mn-2)}$. Letting 
$N=\{(n-1)/r\}$,
$S_1=\{s_1/r\}$, etc., the difference of the numbers of vanishing factors in
the numerator and in the denominator equals
\begin{equation} \label{eq:SO}
\chi\big(r\mid (mn-s_2-s_3-\dots-s_l-1)\big)
-\fl{N-S_1+\frac {1} {r}}-\sum_{k=2}^l\fl{-S_k+\frac {1} {r}}.
\end{equation}
Now there are three cases to distinguish. 

\smallskip
{\sc Case 1: $\fl{N-S_1+\frac {1} {r}}=1$.} 
This means that $N=\frac {r-1} {r}$ and $S_1=0$, that is, 
$r\mid n$ and, in combination with $r\mid (mn-2)$, it follows that
$r=2$. Using \eqref{eq:SA}, we see that $mn-s_2-s_3-\dots-s_l-1=mn-n+s_1$,
whence a further consequence is that $2\mid (mn-s_2-s_3-\dots-s_l-1)$ if,
and only if, $2\mid s_1$. In view of the assumption $S_1=0$, this
implies that $\chi\big(r\mid (mn-s_2-s_3-\dots-s_l-1)\big)=1$.
Since $\fl{-S_k+\frac {1} {r}}=\fl{-S_k+\frac {1} {2}}=0$ for all
$k\ge2$, the sum in \eqref{eq:SO} always equals zero in the current
case. The result of the substitution on the left-hand side of
\eqref{eq:SK} is easily seen to be as asserted on the right-hand side.

\smallskip
{\sc Case 2: $\fl{N-S_1+\frac {1} {r}}=0$.}
Another way to describe the defining condition of this case is
to say that $\{n/r\}\ge\{s_1/r\}$ and not both $r\mid n$ and $r\mid s_1$.
Trivially, we have $-\fl{-S_k+\frac {1}
  {r}}\ge0$. Hence, if $r\mid (mn-s_1-\dots-s_l-1)=(mn-n+s_1)$,
or if $s_k\nequiv 0,1$~(mod~$r$) for at least one $k$ between
$2$ and $l$, then
the sum in \eqref{eq:SO} is positive, and thus the 
left-hand side of \eqref{eq:SK} vanishes.
On the other hand, if $r\nmid (mn-s_1-\dots-s_l-1)=(2-n+s_1)$,
and $s_k\equiv 0,1$~(mod~$r$), $k=2,3,\dots,l$, then \eqref{eq:SO}
equals $0$, and the result of the substitution on the left-hand
side of \eqref{eq:SK} is as asserted on the right-hand side.

\smallskip
{\sc Case 3: $\fl{N-S_1+\frac {1} {r}}=-1$.} 
Since, again, $-\fl{-S_k+\frac {1}
  {r}}\ge0$, the sum in \eqref{eq:SO} is always positive, so
that the left-hand side of \eqref{eq:SK} vanishes.

\medskip
Finally we address \eqref{eq:SL}. As before,
we rewrite the expression on the left-hand
side as product/quotient of factors of the
form $1-q^j$. Subsequently we count how many of these factors 
vanish when $q\to e^{2\pi ip/(mn-2)}$. Letting 
$B_1=\{b_1/r\}$,
$S_1=\{s_1/r\}$, etc., the difference of the numbers of vanishing factors in
the numerator and in the denominator equals
\begin{equation} \label{eq:SP}
\chi\big(r\mid (2-b_1-\dots-b_n)\big)
+\fl{\sum_{k=1}^n B_k-\frac {1} {r}}-\sum_{k=2}^l\fl{-S_k+\frac {1} {r}}.
\end{equation}
Again we have to distinguish three cases. 

\smallskip
{\sc Case 1: $\sum_{k=1}^n B_k=0$.} In other words, we have $B_k=0$
for all $k$, or, equivalently, $r\mid b_k$ for all $k$. 
Because of \eqref{eq:SB}, this implies that also $r\mid n$.
From $r\mid (mn-2)$ it then follows that $r\mid 2$, that is, that
$r=2$. Hence, $\chi\big(r\mid (2-b_1-\dots-b_n)\big)=1$.
Since $\fl{-S_k+\frac {1} {r}}=\fl{-S_k+\frac {1} {2}}=0$, the sum
in \eqref{eq:SP} equals zero. The result of the substitution on 
the left-hand side of \eqref{eq:SL} is as asserted on the
right-hand side.

\smallskip
{\sc Case 2: $1\ge \sum_{k=1}^n B_k\ge\frac {1} {r}$.} In this case,
we have $\fl{\sum_{k=1}^n B_k-\frac {1} {r}}=0$.
If the sum in \eqref{eq:SP} is to vanish, then we must
have $r\nmid (2-b_1-\dots-b_n)$ and $\fl{-S_k+\frac {1} {r}}=0$,
that is, $s_k\equiv0,1$~(mod~$r$) for all $k$ between $2$ and $r$.
As is not difficult to see, the result of the substitution on
the left-hand side of \eqref{eq:SL} is as asserted on the right-hand
side. In all other cases, the result is zero.

\smallskip
{\sc Case 3: $\sum_{k=1}^n B_k\ge1+\frac {1} {r}$.} 
Then $\fl{\sum_{k=1}^n B_k-\frac {1} {r}}=1$, and consequently
the sum in \eqref{eq:SP} is positive, so that the left-hand side
of \eqref{eq:SL} vanishes.

This completes the proof of the lemma.
\end{proof}


\begin{theorem} \label{thm:3}
Let $m,n,s,b_1,b_2,\dots,b_n$ be integers satsifying
the hypotheses in \Cref{lem:1}. 
Furthermore, 
let $C$ be the cyclic group of pseudo-rotations of an $mn$-gon
generated by $\widetilde\phi$. Then
the triple $(M,P,C)$ exhibits the cyclic sieving phenomenon
for the following choices of sets $M$ and polynomials $P$:

\begin{enumerate}
\item 
$M=\widetilde {NC}_+^{(m)}(n)$, and
$P(q)=\frac {1} {[n]_q}\left[\smallmatrix
(m+1)n-2\\n-1\endsmallmatrix\right]_q$;
\item
$M$ consists of all elements of
$\widetilde {NC}_+^{(m)}(n)$ the block sizes of which are all equal to
$m$, and
$P(q)=\frac {1} {[n]_q}\left[\smallmatrix
mn-2\\n-1\endsmallmatrix\right]_q$;
\item
$M$ consists of all elements of
$\widetilde {NC}_+^{(m)}(n)$ which have rank $s$
{\em(}or, equivalently, their number of blocks is $n-s${\em)},
and 
$$P(q)=\frac {1} {[n]_q}
\bmatrix {n}\\{s}\endbmatrix_q
\bmatrix {mn-2}\\ {n-s-1}\endbmatrix_q;
$$
\item
$M$ consists of all elements of
$\widetilde {NC}_+^{(m)}(n)$ whose
number of blocks of size $mi$ is $b_i$, $i=1,2,\dots,n$,
and 
$$P(q)=\frac {1} {[b_1+b_2+\dots+b_n]_q}
\bmatrix {b_1+b_2+\dots+b_n}\\{b_1,b_2,\dots,b_n}\endbmatrix_q
\bmatrix {mn-2}\\ {b_1+b_2+\dots+b_n-1}\endbmatrix_q.
$$
\end{enumerate}
\end{theorem}

\begin{proof}
For Item~(1), this follows by combining \eqref{eq:multichains-pos} and \eqref{eq:SI}.
Item~(2) is the special case of Item~(4) where $b_1=n$
(and, hence, $b_2=b_3=\dots=b_n=0$), and therefore does not need
to be treated separately.

\medskip
We now turn to Item~(3). We need to specialize \eqref{eq:SK} to
the case where $l=2$. Indeed, for this special case \eqref{eq:SK}
reduces to
\begin{multline} \label{eq:ns-siev}
\frac {1} {[n]_q}
\bmatrix {n}\\{s}\endbmatrix_q
\bmatrix {mn-2}\\ {n-s-1}\endbmatrix_q\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}\\
=\begin{cases}
\binom {\fl{(n-1)/r}} {\fl{s/r}}
\binom {(mn-2)/r} {{(n-s-1)/r}},
&\text{if $n-1\equiv s$ \rm (mod $r$)},\\
%&\text{and $r\nmid n$},\\
\binom {(n-2)/2} {{s/2}}
\binom {(mn-2)/2} {(n-s)/2},
&\text{if $r=2$, $n\equiv s\equiv0$ \rm (mod $2$),}\\
0,&\text{otherwise.}
\end{cases}
\end{multline}
Now we have to distinguish two cases, depending on $r$. 

First let $r\ge3$. Then, according to \Cref{lem:10}, any 
positive $m$-divisible non-crossing partition which is
$r$-pseudo-rotationally invariant must have a zero block. 
Moreover, \Cref{lem:12}(1) says that all of these
partitions arise by
Construction~1 and subsequent application of the pseudo-rotation
$\widetilde \phi$. As a consequence,
the number of non-zero blocks must be divisible by $r$.
Consequently, for any such non-crossing partition $\pi$, we have
$$
\#(\text{blocks of $\pi$})=1+rB=n-s,
$$
where $B$ is some non-negative integer. In particular, we must
have $n-1\equiv s$~(mod~$r$).
The number of these non-crossing partitions is given by
Formula~\eqref{eq:multichains-si-pos} with $s$ replaced by
$B=(n-s-1)/r$. It is not difficult to see that this agrees with
the expression in the first option in \eqref{eq:ns-siev}.

Now let $r=2$. In that case, \Cref{lem:12}(2) says that we
obtain all relevant positive $m$-divisible non-crossing partitions
by either Construction~1 or~2 and subsequent application of 
$\widetilde \phi$. In the first case, there is a zero block and
the previous arguments apply again. In the second case, there is
no zero block. Moreover, if $b_i$ denotes the number of blocks
of size $mi$, then all $b_i$'s are even. In particular,
the number of blocks, $n-s$, must be even.
Since $n=b_1+2b_2+\dots+nb_n$, also $n$, and thus also
$s$, must be even. The number of relevant non-crossing partitions
is then given by \eqref{eq:multichains-si-pos-a} with $a=0$ and
$s$ replaced by $(n-s)/2$. The resulting expression is identical
with the expression in the second option in \eqref{eq:ns-siev}.

In all other cases, there are no positive $m$-divisible non-crossing
partitions which are $r$-pseudo-rotationally invariant.

\medskip
Finally, we address Item~(4). Specializing \eqref{eq:SL} to $l=2$,
we obtain
\begin{multline} \label{eq:b-siev}
\frac {1} {[b_1+b_2+\dots+b_n]_q}
\bmatrix {b_1+b_2+\dots+b_n}\\{b_1,b_2,\dots,b_n}\endbmatrix_q
\bmatrix {mn-2}\\ {b_1+b_2+\dots+b_n-1}\endbmatrix_q
\Bigg\vert_{q=e^{2\pi ip/(mn-2)}}\\
=\begin{cases} 
\binom {(b_1+b_2+\dots+b_n-1)/r}
{\fl{b_1/r},\fl{b_2/r},\dots,\fl{b_n/r}}
\binom {(mn-2)/r} {(b_1+b_2+\dots+b_n-1)/r}
&\text{if $b_j\equiv1$ \rm (mod $r$)}\\
&\text{\quad and $b_k\equiv0$ \rm (mod $r$) for $k\ne j$},\\
&\text{\quad for a $j$ between $1$ and $n$},\\
\binom {(b_1+b_2+\dots+b_n)/2}
{{b_1/2},{b_2/2},\dots,{b_n/2}}
\binom {(mn-2)/2} {(b_1+b_2+\dots+b_n)/2}
&\text{if $r=2$ and $n\equiv b_1\equiv\dots\equiv b_n\equiv0$ \rm (mod $2$),}\\
0,&\text{otherwise.}
\end{cases}
\end{multline}

Again, we distinguish two cases depending on $r$.

First let $r\ge3$. Let again $b_i$ denote the number of blocks
of size $mi$.
Repeating the arguments for Item~(3),
we see that there is one $b_j$ (namely the one for which $mj$ is
the size of the zero block) which congruent to $1$ modulo~$r$,
while all other $b_k$'s, $k\ne j$, are divisible by $r$.
The number of relevant non-crossing partitions is then given
by \eqref{eq:multichains-bi-si-pos} with $b_j$ replaced by
$(b_j-1)/r$ and $b_k$ replaced by $b_k/r$, $k\ne j$. 
This immediately yields the expression in the first option 
of \eqref{eq:b-siev}.

Now let $r=2$. In that case, \Cref{lem:12}(2) says that we
obtain all relevant positive $m$-divisible non-crossing partitions
by either Construction~1 or~2 and subsequent application of 
$\widetilde \phi$. In the first case, there is a zero block and
the previous arguments apply again. In the second case, by
repeating the arguments for Item~(3), we see that
all $b_i$'s, and thus also $n$, must be even. 
The number of relevant non-crossing partitions
is then given by \eqref{eq:multichains-bi-si-pos} with 
$b_k$ replaced by $b_k/2$, $k=1,2,\dots,n$. 
The resulting expression is identical
with the expression in second option in \eqref{eq:ns-siev}.

In all other cases, there are no positive $m$-divisible non-crossing
partitions which are $r$-pseudo-rotationally invariant.

\medskip
This completes the proof of the theorem.
\end{proof}

\section{Positive \texorpdfstring{$m$}{m}-divisible non-crossing partitions in type~$B$}
\label{sec:typeB}

In this section, we study the positive Kreweras map in the situation of the 
hyperoctahedral group $B_n$. Recall that $B_n$ can be combinatorially
realized as the group of permutations $\pi$ of
$\{1,2,\dots,n,-1,-2,\dots,-n\}$ satisfying
$\pi(-i)=-\pi(i)$ for $i=1,2,\dots,n$.
In the sequel, we shall most often write $\bar i$ instead of $-i$,
$i=1,2,\dots,n$.
We adopt the type~$B$ cycle notation from \cite{BRWaAA}
consisting of the conventions
$((i_1,i_2,\dots,i_k)):=(i_1,i_2,\dots,i_k)
(\overline{i_1},\overline{i_2},\dots,\overline{i_k})$
and
$[i_1,i_2,\dots,i_k]:=(i_1,i_2,\dots,i_k,
\overline{i_1},\overline{i_2},\dots,\overline{i_k})$.
The Coxeter system corresponding to the hyperoctahedral group $B_n$ is
\begin{equation} \label{eq:WSB} 
(W,\reflS) = \big( B_{n},\{ s_i = ((i,i+1)) : 1 \leq i < n\}\cup
\{s_n=(n,\overline{n})=[n]\} \big)
\end{equation}
together with the Coxeter element $c = s_1 \cdots
s_{n}=[1,2,\ldots,n]$. 
Here, we have the following
combinatorial description of positive $m$-divisible non-crossing
partitions. 
\begin{proposition} \label{lem:wmn>0}
  $(w_0,w_1,\dots,w_m) \in NC^{(m)}(B_n,c)$ is positive if and only if
  $w_m(n)>0$.
\end{proposition}

\begin{proof}
According to \Cref{def:NCpos}, the tuple $(w_0,w_1,\dots,w_m)$ is
positive if and only if $cw_m^{-1}=w_0w_1\cdots w_{m-1}$ does not live
in a proper standard parabolic subgroup of $B_n$, with
respect to our generators $s_i$, $i=1,2,\dots,n$, given in
\eqref{eq:WSB}. The
cycles in the disjoint cycle decomposition of $cw_m^{-1}$ define
a non-crossing partition. Thus, $cw_m^{-1}$ will live in a standard
parabolic subgroup of $B_n$ if and only if the preimage of $1$ under
$cw_m^{-1}$ is a positive number. In other words, there is an $l>0$
such that $(cw_m^{-1})(l)=1$.
By acting on the left on both sides
of this relation by the inverse of the
Coxeter element $c=[1,2,\dots,n]$, we see that this is equivalent
to $w_m^{-1}(l)=\bar n$, or, again equivalently, $w_m(n)=\bar l$,
for some $l>0$. We are interested in the contrapositive:
$cw_m^{-1}$ does {\it not\/} live in a standard parabolic subgroup of
$B_n$ if and only if $w_m(n)>0$.
\end{proof}

Next we recall Armstrong's bijection between $NC^{(m)}(B_{n})$ and 
$m$-divisible non-crossing partitions of
$\{1,2,\dots,mn,\bar1,\bar2,\dots,\overline{mn}\}$ that are 
invariant under a rotation by $180^\circ$, which we denote here by
$\widetilde{NC}{}^{(m)}(B_{n})$.
Given an element $(w_0,w_1,\dots,w_m)\in NC^{(m)}(B_n)$,
the bijection, $\Na^m_{B_n}$ say, 
from \cite[Thm.~4.5.6]{Arm2006} 
works essentially in the same way as in type~$A$: 
namely
$(w_0,w_1,\dots,w_m)\in NC^{(m)}(B_n)$ is mapped to
\begin{equation} \label{eq:Nb}
\Na^m_{B_n}(w_0,w_1,\dots,w_m)=
[1,2,\dots,mn]\,(\bar\ta_{m,1}(w_1))^{-1}\,(\bar\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\bar\ta_{m,m}(w_m))^{-1},
\end{equation}
where $\bar\ta_{m,i}$ is the obvious extension of the 
transformation $\ta_{m,i}$ from \Cref{sec:typeA}.
More precisely, we let
$$(\bar\ta_{m,i}(w))(mk+i-m)=mw(k)+i-m,\quad 
k=1,2,\dots,n,\bar1,\bar2,\dots,\bar n,$$ 
and 
$(\bar\ta_{m,i}(w))(l)=l$ and
$(\bar\ta_{m,i}(w))(\bar l)=\bar l$ for all $l\not\equiv i$~(mod~$m$),
where $m\bar k+i-m$ is identified with $\overline{mk+i-m}$ for all $k$
and $i$. 
Again, the cycles in the disjoint cycle representation correspond to
the blocks in the non-crossing partition in
$\widetilde{NC}{}^{(m)}(B_n)$.
We refer the reader to \cite[Sec.~4.5]{Arm2006} for the details.
For example, let $n=5$, $m=3$, $w_0=((2,4))$,
$w_1=[1]=(1,\bar 1)$,
$w_2=((1,4))$, and
$w_3=((2,3))\,((4,5))$. Then $(w_0;w_1,w_2,w_3)$ is mapped to
\begin{align} \notag
\Na^3_{B_{5}}(w_0;w_1,w_2,w_3)&=[1,2,\dots,15]\,[1]\,((2,11))\,
((6,9))\,((12,15))\\
&=((1,\bar2,\overline{12}))\,((3,4,5,6,10,11))\,((7,8,9))\,((13,14,15)).
\label{eq:NCB}
\end{align} 

\begin{proposition} \label{prop:1B}
The image under $\Na^m_{B_n}$ of the positive $m$-divisible
non-crossing partitions in $NC^{(m)}(B_n)$ are those $m$-divisible
non-crossing partitions in $\widetilde{NC}{}^{(m)}(B_n)$ where 
the block containing $1$ also contains a negative number.
\end{proposition}

\begin{proof}
First, let $(w_0,w_1,\dots,w_m)\in NC^{(m)}(B_n)$.
By definition, we know that $w_m(n)>0$. Let us write $l$
for $w_m(n)$. We then have
\begin{align*}
\big(\Na^m_{B_n}(&w_0,w_1,\dots,w_m)\big)(\overline{ml})\\
&=
\big([1,2,\dots,mn]\,(\ta_{m,1}(w_1))^{-1}\,(\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\ta_{m,m}(w_m))^{-1}\big)(\overline{ml})\\
&=
\big([1,2,\dots,mn]\big)(\overline{mn})=1.
\end{align*}
Translated to non-crossing partitions, this means that $\overline{ml}$ and $1$
belong to the same block, which implies our claim.

Conversely, let $\pi\in \widetilde{NC}{}^{(m)}(B_n)$ be an
$m$-divisible non-crossing partition in which the block containing
$1$ also contains a negative number. Let $\bar L$ be the negative number
that is connected with~$1$ in the geometric representation of $\pi$.
Since in between $\bar L$ and $1$ there ``sit" blocks all of whose sizes
are divisible by $m$, we must necessarily have $\bar L=\overline{ml}$,
for some $l$ between~$1$ and~$n$.
In other words, if $\pi$
is interpreted as a permutation, $\pi(\overline{ml})=1$. Let
$(w_0,w_1,\dots,w_m)$ be the element of $NC^{(m)}(B_n)$ such that
$$
\Na^m_{B_n}(w_0,w_1,\dots,w_m)=\pi.
$$
We must have
$$
\big(\Na^m_{B_n}(w_0,w_1,\dots,w_m)\big)(\overline{ml})=1.
$$
The definition \eqref{eq:Nb} of $\Na^m_{B_n}$
and the fact that $(\ta_{m,i}(w_i))^{-1}$ leaves multiples of $m$
fixed as long as $1\le i\le m-1$ together imply that 
$(\ta_{m,m}(w_m))^{-1}(\overline{ml})=\overline{mn}$.
Phrased differently, we have
$(\ta_{m,m}(w_m))^{-1}(ml)=mn$, or, equivalently, 
$w_m(n)=l>0$. By definition, this means that
$(w_0,w_1,\dots,w_m)$ is positive.
\end{proof} 

The next step consists in translating the positive Kreweras map $\Krewplustilde$
into a ``rotation action" on the positive elements of
$NC^{(m)}(B_n)$. In order to do so, we need to explicitly describe 
the decomposition \eqref{eq:LR} in type~$B$.

\begin{lemma} \label{lem:LRB}
Let~$w$ be an element of ??$NC(B_n,c)$, and consider the cycle $z$
in the (type-$B$) disjoint cycle of~$w$ that contains~$n$. 
\begin{enumerate} 
\item If $z=((i_1,i_2,\dots,i_k,n))$ with $i_1>0$
{\em(}and consequently $0<i_1<i_2<\dots<i_k${\em)}, we put
$a=n$.
\item If $z=((i_1,\dots,i_s,i_{s+1},\dots,i_k,n))$ 
with $i_s<0$ and $i_{s+1}>0$
{\em(}and consequently $i_1,i_2,\dots,i_s<0$,
$i_{s+1},\dots,i_{k-1},i_k>0$,
and $\vert i_1\vert<\vert i_2\vert<\dots<\vert i_k\vert$;
the sequence $i_{s+1},\dots,i_k$ has to be interpreted as
the empty sequence if $s=k${\em)}, 
we put $a=i_s$.
%\item If $z=((i_1,i_2,\dots,i_k,n))$ with $i_k<0$
%{\em(}and consequently $0>i_1>i_2>\dots>i_k${\em)}, we put
%$a=i_k$.
\item If $z=[i_1,i_2,\dots,i_k,n]$
{\em(}and consequently $0<i_1<i_2<\dots<i_k${\em)}, we put
$a=\bar n$.
\end{enumerate}
Then we have
$$w=w^Lw^R,\quad \quad \text {where $w^R=((a,n))$}.$$
{\em(}By slight abuse of notation, here we identify $((\bar n,n))$
with $[n]$.{\em)}
\end{lemma}

\begin{proof}
The last $n$ reflections in $\reflR$ given in \eqref{eq:refls}
are 
\begin{equation} \label{eq:reflsB} 
((i,n)),\ i=\bar1,\bar2,\dots,\overline {n-1},\text{ and }[n]=(\bar n,n),
\end{equation}
in this order. 
Thus, by definition, we have $w^L=w\circ (w^R)^{-1}$, where the factor
$w^R$ consists of a product of these~$n$ reflections (we shall see,
however, that in our case it will be always at most one reflection), 
and $\lenR(w^L)+\lenR(w^R)=\lenR(w)$.
Furthermore, using \Cref{lem:wmn>0} with
$m=1$, we see that $w^L$ must have the property that $w^L(n)>0.$
The arguments below are heavily based on the uniqueness of the
decomposition $w=w^l\circ w^R$ in the sense of 
\Cref{cor:unique}.

In Case~(1), we have $w(n)=i_1>0$. Hence, we may choose
$w^L=w$ and$w^R=\one=((n,n))$, and uniqueness of decomposition
guarantees that this is the correct choice.

In Case~(2), the choice $w^R=((i_s,n))$ implies that
$w^L$ would contain the cycles\break
$((i_1,\dots,i_s))((i_{s+1},\dots,i_k,n))$. 
Hence, we would
have $w^L(n)=i_{s+1}>0$ if~$s<k$, and $w^L(n)=n$ if $s=k$. 
By uniqueness of decomposition, 
this must be the correct choice.

%In Case~(3), the choice $w^R=((i_k,n))$ implies that
%$w^L$ would contain the cycle
%$((i_1,\dots,i_s,i_{s+1},\dots,i_k))$ and map~$n$ to itself. 
%Consequently, this is the correct choice.

Finally we address Case~(3). If we choose $w^R=[n]$,
then $w^L$ would contain the cycle
$((i_1,i_2,\dots,i_k,n))$. 
Hence, we would
have $w^L(n)=i_1>0$. Again,
by uniqueness of decomposition, 
this must be the correct choice.
\end{proof}

Next we have to translate the positive Kreweras map $\Krewplustilde$
into a ``rotation action" on the positive elements of
$NC^{(m)}(B_n)$. 
Let $m\ge2$. We define the map
$\phi:NC^{(m)}_+(W)\to NC^{(m)}_+(W)$ by
\begin{multline} \label{eq:phiB}
(w_0,w_1,\dots,w_m)\\
\mapsto
\big((cw_{m-1}^Rw_mc^{-1})w_0(cw_{m-1}^Rw_mc^{-1})^{-1},
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
\end{multline}
For reasons that will become clear later on,
we shall sometimes call this map {\it pseudo-rotation}.
It is indeed not difficult to see that, if the $(m+1)$-tuple
on the left-hand side is an element of $NC^{(m)}_+(W)$, then so is
the $(m+1)$-tuple on the right-hand side.
Moreover, one can convince oneself (??)
that $\phi^{mn-1}$ acts as the identity.

\begin{remark}
(1) The above definition makes still sense for $m=2$, but it does not for
$m=1$. Whether there is still a sensible way to define $\phi$ even
for $m=1$ remains to be seen.
We shall see that, at least for the types $A_n$, $B_n$, and $D_n$
there are meaningful?? extensions which also cover the case where $m=1$,
see Definitions~??.

\medskip
(2) If $w_0=\one$, then the action $\phi$ reduces to
$$
(\one,w_1,\dots,w_m)
\mapsto
\big(\one,
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
$$
\end{remark}

\begin{proposition} \label{prop:2B}
Let $m\ge2$.
Under the bijection $\Na^m_{B_n}$, the map $\phi$ translates to
the following map $\widetilde\phi$ on $\widetilde{NC}{}^{(m)}_+(B_n)$:
if the block of $\overline{mn}$ contains another negative element,
then $\phi$ rotates all blocks of $\pi$ by one unit in clockwise direction.

On the other hand, 
if the block of $\overline{mn}$ of some positive $m$-divisible
non-crossing partition~$\pi$ 
of $\widetilde{NC}{}^{(m)}_+(B_n)$ contains no other negative 
element, then, first of all, $1$ is also contained in this block.
In other words, $\overline mn$ and $1$ are successive elements
in a block {\em(}and consequently also $mn$ and $\bar 1${\em)}.
For the description of the operation, we have to distinguish 
between two cases:

\begin{enumerate} 
\item If there is a block not containing $1,\bar1,mn,\overline{mn}$ in
  which there are positive {\em and} negative elements, then
let $b$ be minimal such that $\bar d$ and $b$ are successive
elements in a block, with $\bar d$ negative and $b$ positive.
Furthermore, let $b$ be followed by $e$ in this block, and let
$a$ and $\overline{mn}$ be successive elements
in a block. {\em(}The element~$a$ must be positive by the assumption
defining the subcase in which we are, while $e$ can be positive
or negative. See the left half of \Cref{fig:10} for an
illustration of the various definitions.{\em)} 
Then the image of $\pi$ under $\widetilde\phi$ is the partition 
in which $\overline{d+1}$, $1$, and $e+1$ are three 
successive elements in a
block, and $a+1$, $b+1$, and $2$ are successive elements in 
another block. All other succession relations in $\pi$
are rotated by one unit in clockwise direction.
See \Cref{fig:10} for a schematic illustration of this operation.
\item If there is {\em no} block other than the ones containing
$1,\bar1,mn,\overline{mn}$ with
positive {\em and} negative elements, then let $a$,
$\overline{mn}$, and~$1$ be successive elements in a block.
{\em(}Again, the element~$a$ must be positive.{\em)}
The image of $\pi$ under $\widetilde\phi$ is the partition 
in which $\overline{a+1}$, $1$, and $\bar2$ are successive
elements. See \Cref{fig:11} for a schematic illustration of this
operation.
\end{enumerate}
\end{proposition}

\begin{figure}
\begin{center}
  \begin{tikzpicture}[scale=1]
    \polygon{(-4,0)}{obj}{44}{2.5}
       {1,,,,a,,,,,,b,,,,e,,,d,,,,mn,\overline{1},,,,\overline{a},,,,,,\overline{b},,,,\overline{e},,,\overline{d},,,,\overline{mn}}

%     thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj1) to[bend right=50]
(obj5);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj23) to[bend right=50]
(obj27);

    % line connecting a and \overline{mn}
    \draw (obj44) to[bend right=50] (obj5);
    \draw (obj22) to[bend right=50] (obj27);

    % area for a+1 ... b-1
    \draw[dotted] (obj6) to[bend right=50] (obj10);
    \node at ($($0.92*($(obj8)+(4.0,0)$)$)-(4.0,0)$) {\tiny$X$};

    \draw[dotted] (obj28) to[bend right=50] (obj32);
    \node at ($($0.92*($(obj30)+(4.0,0)$)$)-(4.0,0)$) {\tiny$\overline{X}$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj40) to[bend right=20] (obj11) to[bend right=50] (obj15);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj15) to[bend left=10] (obj40);
    \draw (obj18) to[bend right=20] (obj33) to[bend right=50] (obj37);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj37) to[bend left=10] (obj18);

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{2.5}
       {1,2,,,,a+1,,,,,,b+1,,,,e+1,,,d+1,,,mn,\overline{1},\overline{2},,,,\overline{a+1},,,,,,\overline{b+1},,,,\overline{e+1},,,\overline{d+1},,,\overline{mn}}

%     thick line connecting 1 and \overline{d+1}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj41);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj19);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj2) to[bend right=50]
(obj6);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj24) to[bend right=50]
(obj28);

    % line connecting a+1 and b
    \draw (obj6) to[bend right=50] (obj12);
    \draw (obj28) to[bend right=50] (obj34);

    % line connecting b+1 and 2
    \draw (obj2) to[bend right=50] (obj12);
    \draw (obj24) to[bend right=50] (obj34);

    % area for a+2 ... b
    \draw[dotted] (obj7) to[bend right=50] (obj11);
    \node at ($($0.92*($(obj9)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj29) to[bend right=50] (obj33);
    \node at ($($0.92*($(obj31)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj1) to[bend right=20] (obj16);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj16) to[bend left=10] (obj41);
    \draw (obj23) to[bend right=20] (obj38);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj38) to[bend left=10] (obj19);

    \end{tikzpicture}
\end{center}
\caption{The action of the pseudo-rotation $\widetilde\phi$,
Case (1)}
\label{fig:10}
\end{figure}

\begin{figure}
\begin{center}
  \begin{tikzpicture}[scale=1]
    \polygon{(-4,0)}{obj}{44}{2.5}
       {1,,,,a,,,,,,,,,,,,,,,,\hspace{20pt}mn-1,mn,\overline{1},,,,\overline{a},,,,,,,,,,,,,,,,\overline{mn-1}\hspace{20pt},\overline{mn}}

%     thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj1) to[bend right=50]
(obj5);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj23) to[bend right=50]
(obj27);

    % line connecting a and \overline{mn}
    \draw (obj44) to[bend right=50] (obj5);
    \draw (obj22) to[bend right=50] (obj27);

    % area for a+1 ... mn-1
    \draw[dotted] (obj6) to[bend right=20] (obj21);
    \node at ($($0.62*($(obj13)+(4.0,0)$)$)-(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj28) to[bend right=20] (obj43);
    \node at ($($0.62*($(obj35)+(4.0,0)$)$)-(4.0,0)$) {\tiny$\overline{X}$};

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{2.5}
       {1,2,,,,a+1,,,,,,,,,,,,,,,,mn,\overline{1},\overline{2},,,,\overline{a+1},,,,,,,,,,,,,,,,\overline{mn}}

%     thick line connecting 1 and \overline{a+1}
    \draw[line width=2.5pt,black] (obj1) to[bend left=20] (obj28);
    \draw[line width=2.5pt,black] (obj23) to[bend left=20] (obj6);

    % dotted line connecting 1 and \overline{2}
    \draw (obj1) to[bend right=0] (obj24);
    \draw (obj23) to[bend right=0] (obj2);

    % dotted line connecting 2 and a+1
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj2) to[bend right=50]
(obj6);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj24) to[bend right=50]
(obj28);

    % area for a+2 ... mn
    \draw[dotted] (obj7) to[bend right=20] (obj22);
    \node at ($($0.62*($(obj14)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj29) to[bend right=20] (obj44);
    \node at ($($0.62*($(obj36)-(4.0,0)$)$)+(4.0,0)$) {\tiny$\overline{X}$};

    \end{tikzpicture}
\end{center}
\caption{The action of the pseudo-rotation $\widetilde\phi$,
Case (2)}
\label{fig:11}
\end{figure}


\begin{remark}
??\Cref{fig:10} provides a schematic illustration of 
the construction in the statement of 
the above proposition, while \Cref{fig:11} provides a concrete
example. In this example, $m=3$, $n=8$, $a=14$,
$b_1=15$, and $b_2=19$.
\end{remark}

\begin{proof}[Proof of \Cref{prop:2B}]
Let us first assume that, 
in the $m$-divisible type-$B$ non-crossing partition~$\pi$,
the block of $\overline{mn}$ contains another negative element.
Among those, let $\overline{mi_1-1}$ be the one which is the
predecessor of $\overline{mn}$ in the block.
Obviously, we have $i_1>0$.
\Cref{fig:16} provides a sketch of this situation. 
Let $(w_0,w_1,\dots,w_m)$ denote the
element of $NC^{(m)}_+(B_n)$ which corresponds to $\pi$ under the
bijection $\Na^m_{B_n}$. By the definition of $\Na^m_{B_n}$, we
see that $w_{m-1}(n)=i_1$. From
\Cref{lem:LRB}.(1), we infer $w_{m-1}^L=w_{m-1}$ and
$w_{m-1}^R=\one$. This means that the pseudo-rotation \eqref{eq:phiB}
reduces to ordinary rotation in this special case, as we claimed. 

\begin{figure}
??
  \caption{The elements $\overline{mn}$ and $1$ are not in the same block}
\label{fig:16}
\end{figure}

\medskip
For the remainder of the proof, we assume that the block of
$\overline{mn}$ contains no other negative element.
We first show that also~$1$ is an element of that block.
To see this, let $a$ be the ekement before $\overline{mn}$
in the block of $\overline{mn}$. By assumption, $\overline{mn}$ is the
only negative element in its block. Consequently, $a$ is positive.
On the other hand, by \Cref{prop:1B}, we know that
the block of $1$ contains at least one negative element.
If $1$ would not be in the same block as $\overline{mn}$,
then we would obtain a contradiction, since the ``block connection"
between $a$ and $\overline{mn}$ would prevent $1$ to be connected
to any negative element. Hence, $1$ must indeed be in the same
block as $\overline{mn}$.

\medskip
Let again $(w_0,w_1,\dots,w_m)$ denote the
element of $NC^{(m)}_+(B_n)$ which corresponds to $\pi$ under the
bijection $\Na^m_{B_n}$. 

\smallskip
{\sc Case (1).} 
Let $a_2,a_3,\dots,a_{s-1}$ be positive integers with
$a<a_2<a_3<\dots<a_{s-1}$ such that 
$a_2$ and $a+1$ are successive elements in a block,
$a_3$ and $a_2+1$ are successive elements in a block,
\dots, and
$b-1$ and $a_{s-1}+1$ are successive elements in a block.
Figure~?? provides a sketch of this situation.

The congruence classes of $a,b,d,e,a_2,\dots,a_{s-1}$ 
modulo~$m$ are forced: indeed, 
we have $a\equiv-1$~(mod~$m$) since $a$ is the predecessor of
$\overline{mn}$ in their block,
we have $a_2\equiv-1$~(mod~$m$) since $a_2$ is the predecessor of
$a+1$ in their block,
we have $a_3\equiv-1$~(mod~$m$) since $a_3$ is the predecessor of
$a_2+1$ in their block, \dots,
we have $b\equiv0$~(mod~$m$) since $b-1$ is the predecessor of
$a_{s-1}+1$ in their block, and
we have $d\equiv-1$~(mod~$m$) and
$e\equiv1$~(mod~$m$) since $\bar d$, $b$, $e$ are successive
elements in their block. We write
$a=mA-1$, 
$b=mB$,
$d=mD-1$, 
$e=mE+1$, and $a_i=mA_i-1$, $i=2,3,\dots,s-1$, 
for suitable integers 
$A,B,D,E,A_2,A_3,\dots,A_{s-1}$. 
By the definition of  $\Na^m_{B_n}$,
we have
\begin{align}
\notag
%\label{eq:wmex1}
w_{m-1}(\bar n)&=A,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A)&=A_2,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A_2)&=A_3,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A_{s-1})&=B,\\
\notag
%\label{eq:wmex2}
w_{m-1}(B)&=\bar D,\\
\label{eq:wmex3}
w_m(E)&=B,\\
\label{eq:wmex4}
w_m(n)&=n.
\end{align}

\begin{figure}
??
  \caption{Illustration of the meaning of $a,b,d,e$}
\label{fig:15}
\end{figure}

\noindent
Consequently, the cycle of $w_{m-1}$ containing $n$ is of the
form 
$$((\bar A,\bar A_2,\dots,\bar A_{s-1},\bar B,D,\dots,n)).$$
By \Cref{lem:LRB}.(2), we have 
$$w^R_{m-1}=((\bar B,n)),\quad \text{and }
w^L_{m-1}\text{ contains the cycles }((A,A_2,\dots,A_{s-1},B))
((D,\dots,n)).$$
Together with \eqref{eq:wmex3} and \eqref{eq:wmex4}, 
this implies the relations
\begin{align*}
%\label{eq:wmex1}
w^L_{m-1}(n)&=D,\\
%\label{eq:wmex1}
w^L_{m-1}(A)&=A_2,\\
%\label{eq:wmex1}
w^L_{m-1}(A_2)&=A_3,\\
%\label{eq:wmex1}
w^L_{m-1}(A_{s-1})&=B,\\
%\label{eq:wmex2}
w^L_{m-1}(B)&=A,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(1)&=B+1,\\
%\label{eq:wmex4}
(cw^R_{m-1}w_mc^{-1})(E+1)&=1.
\end{align*}
By the definition of  $\Na^m_{B_n}$, this means that
$a+1$, $b+1$, and~$2$ are successive elements in their block,
that $\overline{d+1}$, $1$, and $e+1$ are successive elements
in a block,
while all other ``block connections" are rotated by one unit.
This is in accordance with the corresponding assertion in the
proposition.

\smallskip
{\sc Case (2).} 
Let $a_2,a_3,\dots,a_k$ be positive integers with
$a<a_2<a_3<\dots<a_k$ such that 
$a_2$ and $a+1$ are successive elements in a block,
$a_{i+1}$ and $a_i+1$ are successive elements in a block,
$i=2,3,\dots,k-1$, and
$mn-1$ and $a_k+1$ are successive elements in a block.
Figure~?? provides a sketch of this situation.

As in Case~(1), we must have $a\equiv-1$~(mod~$m$),
and $a_i\equiv-1$~(mod~$m$), for $i=2,3,\dots,k$.
We write again $a=mA-1$ and $a_i=mA_i-1$ for suitable 
integers $A$ and $A_i$, for $i=2,3,\dots,k$.
By the definition of  $\Na^m_{B_n}$,
we have
\begin{align}
\notag
%\label{eq:wmex1}
w_{m-1}(\bar n)&=A,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A)&=A_2,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A_2)&=A_3,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A_{k-1})&=A_k,\\
\notag
%\label{eq:wmex1}
w_{m-1}(A_k)&=n,\\
\label{eq:wmex4b}
w_m(n)&=n.
\end{align}
Consequently, the cycle of $w_{m-1}$ containing $n$ is of the
form 
$$[A,A_2,\dots,\dots,A_k,n].$$
By \Cref{lem:LRB}.(3), we have 
$$w^R_{m-1}=[n],\quad \text{and }
w^L_{m-1}\text{ contains the cycle }((A,A_2,\dots,A_k,n)).$$
Together with \eqref{eq:wmex4b}, 
this implies the relations
\begin{align*}
%\label{eq:wmex1}
w^L_{m-1}(n)&=A,\\
%\label{eq:wmex1}
w^L_{m-1}(A)&=A_2,\\
%\label{eq:wmex1}
w^L_{m-1}(A_2)&=A_3,\\
%\label{eq:wmex1}
w^L_{m-1}(A_{k-1})&=A_k,\\
%\label{eq:wmex2}
w^L_{m-1}(A_k)&=n,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(1)&=\bar 1.
\end{align*}
By the definition of  $\Na^m_{B_n}$, this means that
$\overline{a+1}$, $1$, and~$\bar 2$ are successive elements in 
their block,
while all other ``block connections" are rotated by one unit.
This is in accordance with the corresponding assertion in the
proposition.

This completes the proof of the proposition.
\end{proof}


??general definition of map $\widetilde\phi$, also for $m=1$

\begin{lemma} \label{lem:N-2B}
The order of the map $\widetilde\phi$ is $N-1$.
\end{lemma}

\begin{proof}
??
\end{proof}


\subsection{Enumeration}
\label{sec:enumB}

\subsection{Pseudo-rotationally invariant elements and cyclic sieving}
\label{sec:rotB}

\medskip
\noindent
{\sc Construction.} 
We start with an (ordinary)
??$r$-rotationally invariant 
non-crossing partition of $\{2,3,\dots,mn,\bar
2,\bar2,\dots,\overline{mn}\}$,
all of whose non-zero blocks have size divisible by $m$, $mn$ being
contained in the zero block. 
Then we add $1$ and $\bar1$ to the zero block, thereby creating a
positive $m$-divisible non-crossing partition.
\Cref{fig:3,fig:4}?? illustrate this construction.
The choice of parameters in these figures is $m=3$ and $n=10$.
In order to obtain the non-crossing partition in \Cref{fig:3},
one starts with the non-crossing partition 
\begin{multline*}
\{\{2,9,16,23\},\{3,7,8\},\{4,5,6\},
\{10,14,15\},\{11,12,13\},\\
\{17,21,22\},\{18,19,20\},
\{24,28,29\},\{25,26,27\}\}
\end{multline*}
of $\{2,3,\dots,\overline{30}\}$, 
which is indeed $4$-rotationally invariant
in the above sense. Then one adds $1$ and $\bar1$ to the
zero block $\{2,9,16,23\}$ to obtain the partition shown in
\Cref{fig:3}. If one applies $\widetilde\phi$ twice to this
partition, then the result is the partition in \Cref{fig:4}.

\begin{lemma} \label{lem:13}
If $r\ge2$??,
all $r$-pseudo-rotationally invariant positive $m$-divisible non-crossing
  partitions of $\{1,2,\dots,mn,\bar1,\bar2,\dots,\overline{mn}\}$
??type~$B$ 
are obtained by starting with a non-crossing partition
obtained by the above Construction
and applying the pseudo-rotation $\widetilde\phi$ repeatedly to it.
\end{lemma}


\subsubsection*{Cyclic sieving}

\section{Positive \texorpdfstring{$m$}{m}-divisible non-crossing partitions in type~$D$}
\label{sec:typeD}

In this section, we study the positive Kreweras map in the situation of the 
group $D_n$. Recall that $D_n$ can be combinatorially
realized as the group of permutations $\pi$ of
$\{1,2,\dots,n,-1,-2,\dots,-n\}$ satisfying
$\pi(-i)=-\pi(i)$ for $i=1,2,\dots,n$ which maps an even number
of the numbers $\{1,2,\dots,n\}$ to a negative number.
Again, we shall most often write $\bar i$ instead of $-i$,
$i=1,2,\dots,n$. Moreover, the type~$B$ cycle notation explained
in \Cref{sec:typeB} is also used here.
The Coxeter system corresponding to the hyperoctahedral group $D_n$ is
\begin{equation} \label{eq:WSD} 
(W,\reflS) = \big( D_{n},\{ s_i = ((i,i+1)) : 1 \leq i \le n-1\}\cup
\{s_{n}=((n-1,\bar n))\} \big)
\end{equation}
together with the Coxeter element $c = s_1 \cdots
s_{n}=[1,2,\ldots,n-1][n]$. 
Here, we have the following
combinatorial description of positive $m$-divisible non-crossing
partitions. 
\begin{proposition} \label{lem:wmn-1>0}
  $(w_0,w_1,\dots,w_m) \in NC^{(m)}(D_n,c)$ is positive if and only if\break
  $0<w_m(n-1)\le n-1$.
\end{proposition}

\begin{proof}
According to Definition~\ref{def:NCpos}, the tuple $(w_0,w_1,\dots,w_m)$ is
positive if and only if $cw_m^{-1}=w_0w_1\cdots w_{m-1}$ does not live
in a proper standard parabolic subgroup of $D_n$, with
respect to our generators $s_i$, $i=1,2,\dots,n$, given in
\eqref{eq:WSD}. The
cycles in the disjoint cycle decomposition of $cw_m^{-1}$ define
a non-crossing partition. Thus, $cw_m^{-1}$ will live in a standard
parabolic subgroup of $D_n$ if and only if the preimage of $1$ under
$cw_m^{-1}$ is a positive number or $\bar n$. 
In other words, there is either an $l>0$
such that $(cw_m^{-1})(l)=1$, or $(cw_m^{-1})(\bar n)=1$.
By acting on the left on both sides
of this relation by the inverse of the
Coxeter element $c=[1,2,\dots,n-1][n]$, we see that this is equivalent
to either $w_m^{-1}(l)=\bar {n-1}$, or $w_m^{-1}(\bar n)=\bar {n-1}$.
Again an equivalent statement is that we have either $w_m(n-1)=\bar
l$, for some $l>0$,
or $w_m(n-1)=n$. We are interested in the contrapositive:
$cw_m^{-1}$ does {\it not\/} live in a standard parabolic subgroup of
$D_n$ if and only if $0<w_m(n-1)\le n-1$.
\end{proof}

Next we recall the bijection from
\cite[Sec.~7]{KrMuAB} between $NC^{(m)}(D_{n})$ and 
$m$-divisible non-crossing partitions on an annulus, with
the numbers
$\{1,2,\dots,mn-m,\bar1,\bar2,\dots,\break\overline{mn-m}\}$ clockwise
around the outer circle, the numbers 
$\{mn-m+1,mn-m+2,\dots,mn,\break
\overline{mn-m+1},\overline{mn-m+2},\dots,\overline{mn}\}$ 
counter-clockwise around the inner circle, 
that are invariant under a rotation by $180^\circ$, 
and which satisfy the additional
restriction that successive elements in a block must be
successive numbers when taken modulo~$m$. 
In the case that there is no block that contains elements 
from the outer {\it and\/} the inner circle, the blocks
on the inner circle are completely determined by the blocks
on the outer circle in the following way: consider a block, $B$ say,
on the outer circle that is ``visible" from the inner circle.
(Here, ``visible" means that it is possible to travel from
a point of the inner circle to an element of that block
without crossing any block connection.) Let $b$ be the ``right-most"
element in $B$, which, by definition, is determined by considering
the elements of $B\cup(-B)$ in clockwise order and, among these,
choosing the element of $B$ that is followed by an element of
$-B$. There is a unique $a\equiv b$~(mod~$m$) between
$mn-m+1$ and $mn$. Then one block on the inner circle
consists of the $m$ consecutive elements in counter-clockwise 
order on the inner circle ending in $a$, and the other
block consists of the negatives of these elements.
(This last condition was missing
in \cite{KrMuAB}, but is reported in \cite[Sec.~7]{JKimAA}.)
We denote these latter $m$-divisible non-crossing partitions by
$\widetilde{NC}{}^{(m)}(D_{n})$.

Given an element $(w_0,w_1,\dots,w_m)\in NC^{(m)}(D_n)$,
the bijection, $\Na^m_{D_n}$ say, 
from \cite[Sec.~7]{KrMuAB} 
works essentially in the same way as in type~$B$: 
namely
$(w_0,w_1,\dots,w_m)\in NC^{(m)}(D_n)$ is mapped to
\begin{multline} \label{eq:Nd}
\Na^m_{D_n}(w_0,w_1,\dots,w_m)=
[1,2,\dots,m(n-1)]\,[mn-m+1,\dots,mn-1,mn]\\
\circ(\bar\ta_{m,1}(w_1))^{-1}\,(\bar\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\bar\ta_{m,m}(w_m))^{-1},
\end{multline}
where $\bar\ta_{m,i}$ is defined as in \Cref{sec:typeB}
for type~$B$.
Again, the cycles in the disjoint cycle representation correspond to
the blocks in the non-crossing partition in
$\widetilde{NC}{}^{(m)}(D_n)$.
We refer the reader to \cite[Sec.~7]{KrMuAB} for the details.
For example, let $n=6$, $m=3$, $w_0=((2,\bar 4))$,
$w_1=((2,\bar 6))\,((4,5))$,
$w_2=((1,\bar 5))\,((2,3))$, and
$w_3=((3,6))$. Then $(w_0;w_1,w_2,w_3)$ is mapped to
\begin{align} \notag
\Na^3_{D_{6}}(w_0;&w_1,w_2,w_3)\\
\notag
&=[1,2,\dots,15]\,[16,17,18]
((4,\overline{16}))\,((10,13))\,((2,\overline{14}))\,
((5,8))\,((9,18))\\
&=((1,2,\overline{15}))\,((3,4,\overline{17},\overline{18},\overline{10},
\overline{14}))\,((5,9,\overline{16}))\,((6,7,8))\,((11,12,13)).
\label{eq:NCD}
\end{align} 

\begin{proposition} \label{prop:1D}
The image under $\Na^m_{D_n}$ of the positive $m$-divisible
non-crossing partitions in $NC^{(m)}(D_n)$ are those $m$-divisible
non-crossing partitions in $\widetilde{NC}{}^{(m)}(D_n)$ where 
the predecessor of $1$ in its block is a negative number 
on the outer circle of the annulus.
\end{proposition}

\begin{proof}
First, let $(w_0,w_1,\dots,w_m)\in NC^{(m)}(D_n)$.
By definition, we know that\break $0<w_m(n-1)\le n-1$. Let us write $l$
for $w_m(n-1)$. We then have
\begin{align*}
\big(\Na^m_{D_n}(&w_0,w_1,\dots,w_m)\big)(\overline{ml})\\
&=
\big([1,2,\dots,mn-m][mn-m+1,\dots,mn]\\
&\kern2cm
\,(\ta_{m,1}(w_1))^{-1}\,(\ta_{m,2}(w_2))^{-1}\,\cdots\,
(\ta_{m,m}(w_m))^{-1}\big)(\overline{ml})\\
&=
\big([1,2,\dots,mn-m][mn-m+1,\dots,mn]\big)(\overline{m(n-1)})=1.
\end{align*}
Translated to non-crossing partitions, this means that $\overline{ml}$ and $1$
belong to the same block, which implies our claim.

Conversely, let $\pi\in \widetilde{NC}{}^{(m)}(D_n)$ be an
$m$-divisible non-crossing partition in which 
$1$ is preceded in its block by a negative number,
$\bar L$ say, on the outer circle.
Since in between $\bar L$ and $1$ there ``sit" blocks all of whose
sizes 
are divisible by $m$, we must necessarily have $\bar L=\overline{ml}$,
for some $l$ between~$1$ and~$n-1$.
In other words, if $\pi$
is interpreted as a permutation, $\pi(\overline{ml})=1$. Let
$(w_0,w_1,\dots,w_m)$ be the element of $NC^{(m)}(D_n)$ such that
$$
\Na^m_{D_n}(w_0,w_1,\dots,w_m)=\pi.
$$
We must have
$$
\big(\Na^m_{D_n}(w_0,w_1,\dots,w_m)\big)(\overline{ml})=1.
$$
The definition \eqref{eq:Nd} of $\Na^m_{D_n}$
and the fact that $(\ta_{m,i}(w_i))^{-1}$ leaves multiples of $m$
fixed as long as $1\le i\le m-1$ together imply that 
$(\ta_{m,m}(w_m))^{-1}(\overline{ml})=\overline{m(n-1)}$.
Phrased differently, we have
$(\ta_{m,m}(w_m))^{-1}(ml)=m(n-1)$, or, equivalently, 
$w_m(n-1)=l>0$. By definition, this means that
$(w_0,w_1,\dots,w_m)$ is positive.
\end{proof} 

The next step consists in translating the positive Kreweras map $\Krewplustilde$
into a ``rotation action" on the positive elements of
$NC^{(m)}(D_n)$. In order to do so, we need to explicitly describe 
the decomposition \eqref{eq:LR} in type~$D$.

\begin{lemma} \label{lem:LRD}
Let~$w$ be an element of ??$NC(D_n,c)$, and consider the cycle $z$
in the (type-$D$) disjoint cycle of~$w$ that contains~$n-1$. 
\begin{enumerate} 
\item If $z=((i_1,i_2,\dots,i_k,n-1))$ with 
$0<i_1<i_2<\dots<i_k$, then
$w^R=\one$.
\item If $z=((i_1,\dots,i_s,i_{s+1},\dots,i_k,n-1))$ 
with $i_1,i_2,\dots,i_s<0$,
$i_{s+1},\dots,i_{k-1},i_k>0$,
and $\vert i_1\vert<\vert i_2\vert<\dots<\vert i_k\vert$
{\em(}as before, 
the sequence $i_{s+1},\dots,i_k$ has to be interpreted as
the empty sequence if $s=k${\em)}, 
then $w^R=((i_s,n-1))$.
\item If $z=((i_1,\dots,i_s,n-1,j_1,\dots,j_t,\pm n))$ 
with $0<i_1<i_2<\dots<i_s<n-1$ and
$0>j_1>j_2>\dots>j_t>-i_1$, 
then $w^R=((n-1,\pm n))$.
\item If $z=[i_1,i_2,\dots,i_k,n-1][n]$
{\em(}and consequently $0<i_1<i_2<\dots<i_k${\em)}, then
$w^R=((n-1,n))((n-1,\bar n))=[n-1][n]$.
\end{enumerate}
Here, in Item~{\em(3)}, $\pm n$ means either $n$ or $\bar n$,
and the two occurrences must be read consistently.
\end{lemma}

\begin{proof}
The last $n$ reflections in $\reflR$ given in \eqref{eq:refls}
are 
\begin{equation} \label{eq:reflsD} 
((i,n-1)),\ i=\bar1,\bar2,\dots,\overline {n-2},\text{ and }
((n-1,\bar n)),\ ((n-1,n)),
\end{equation}
in this order. 
Thus, by definition, we have $w^L=w\circ (w^R)^{-1}$, where the factor
$w^R$ consists of a product of these~$n$ reflections, 
and $\lenR(w^L)+\lenR(w^R)=\lenR(w)$.
Furthermore, using \Cref{lem:wmn-1>0} with
$m=1$, we see that $w^L$ must have the property that 
$0\le w^L(n-1)\le n-1.$
The arguments below are heavily based on the uniqueness of the
decomposition $w=w^l\circ w^R$ in the sense of 
\Cref{cor:unique}.

In Case~(1), we have $w(n-1)=i_1>0$. Hence, we may choose
$w^L=w$ and$w^R=\one$, and uniqueness of decomposition
guarantees that this is the correct choice.

In Case~(2), the choice $w^R=((i_s,n-1))$ implies that
$w^L$ would contain the cycles
$((i_1,\dots,i_s))((i_{s+1},\dots,i_k,n-1))$. 
Hence, we would
have $w^L(n-1)=i_{s+1}>0$ if~$s<k$, and $w^L(n-1)=n-1$ if $s=k$. 
By uniqueness of decomposition, 
this must be the correct choice.

Case~(3) is similar. The choice $w^R=((n-1,\pm n))$ 
implies that
$w^L$ would contain the cycles
$((i_1,i_2,\dots,i_s,n-1))((j_1,j_2,\dots,j_t,\pm n))$. 
Hence, we would
have $w^L(n-1)=i_1>0$, which,
by uniqueness of decomposition, 
must be the correct choice.

Finally we address Case~(4). If we choose $w^R=[n-1][n]$,
then $w^L$ would contain the cycle
$((i_1,i_2,\dots,i_k,n-1))$. 
Hence, we would
have $w^L(n-1)=i_1>0$. Again,
by uniqueness of decomposition, 
this must be the correct choice.
\end{proof}

Next we have to translate the positive Kreweras map $\Krewplustilde$
into a ``rotation action" on the positive elements of
$NC^{(m)}(D_n)$. 
Let $m\ge2$. We define the map
$\phi:NC^{(m)}_+(W)\to NC^{(m)}_+(W)$ by
\begin{multline} \label{eq:phiD}
(w_0,w_1,\dots,w_m)\\
\mapsto
\big((cw_{m-1}^Rw_mc^{-1})w_0(cw_{m-1}^Rw_mc^{-1})^{-1},
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
\end{multline}
For reasons that will become clear later on,
we shall sometimes call this map {\it pseudo-rotation}.
It is indeed not difficult to see that, if the $(m+1)$-tuple
on the left-hand side is an element of $NC^{(m)}_+(W)$, then so is
the $(m+1)$-tuple on the right-hand side.
Moreover, one can convince oneself (??)
that $\phi^{2m(n-1)-2}$ acts as the identity.

\begin{remark}
(1) The above definition makes still sense for $m=2$, but it does not for
$m=1$. Whether there is still a sensible way to define $\phi$ even
for $m=1$ remains to be seen.
We shall see that, at least for the types $A_n$, $B_n$, and $D_n$
there are meaningful?? extensions which also cover the case where $m=1$,
see Definitions~??.

\medskip
(2) If $w_0=\one$, then the action $\phi$ reduces to
$$
(\one,w_1,\dots,w_m)
\mapsto
\big(\one,
cw_{m-1}^Rw_mc^{-1},w_1,w_2,\dots,w_{m-1}^L\big).
$$
\end{remark}

\begin{proposition} \label{prop:2D}
Let $m\ge2$.
Under the bijection $\Na^m_{D_n}$, the map $\phi$ translates to
the following map $\widetilde\phi$ on $\widetilde{NC}{}^{(m)}_+(D_n)$:
if the block of $\overline{m(n-1)}$ contains another negative element
which is different from $\overline{mn-1}$,
then $\phi$ rotates all blocks of $\pi$ by one unit
{\em(}in clockwise direction on the outer circle of the annulus,
and in counter-clockwise direction on the inner circle{\em)}.

On the other hand, 
if the block of $\overline{m(n-1)}$ of some positive $m$-divisible
non-crossing partition~$\pi$ 
of $\widetilde{NC}{}^{(m)}_+(D_n)$ contains no other negative 
element which is different from $\overline{mn-1}$, 
then, first of all, $1$ is also contained in this block.
In other words, $\overline{m(n-1)}$ and\/ $1$ are successive elements
in a block {\em(}and consequently also $m(n-1)$ and\/ $\bar 1${\em)}.
For the description of the operation, we have to distinguish 
between three cases:

\begin{enumerate} 
\item If there is a block not containing 
$1,\bar1,m(n-1),\overline{m(n-1)}$ in
  which there are positive {\em and} negative elements
from the outer circle, then
let $b$ be minimal such that $\bar d$ and $b$ are successive
elements in such a block, 
with $\bar d$ negative and $b$ positive.
Furthermore, let $b$ be followed by $e$ in this block, and let
$a$ and $\overline{m(n-1)}$ be successive elements
in a block. {\em(}The element~$a$ must be positive and on the
outer circle by the assumption
defining the subcase in which we are, while $e$ can be positive
or negative, and on the outer or inner circle. See 
the left half of \Cref{fig:12} for an
illustration of the various definitions.{\em)} 
Then the image of $\pi$ under $\widetilde\phi$ is the partition 
in which $\overline{d+1}$, $1$, and $e+1$ are three 
successive elements in a
block, and $a+1$, $b+1$, and $2$ are successive elements in 
another block. All other succession relations in $\pi$
are rotated by one unit in clockwise direction.
See \Cref{fig:12} for a schematic illustration of this operation.
\item If the predecessor of $\mp mn$, denoted by $\bar d$, 
is on the outer circle, then let 
$\bar b$ be the successor of $\mp mn$ and
$a$ be the predecessor of
$\overline{m(n-1)}$.
{\em(}The element~$a$ must be positive and on the
outer circle or equal to $\mp(mn-1)$ by the assumption
defining the subcase in which we are. Furthermore,
the element $\bar b$ is either a negative element on the
outer circle or equal to $\pm(mn-m+1)$.
See the left half of \Cref{fig:13} for an
illustration of the described situation.{\em)} 
Then the image of $\pi$ under $\widetilde\phi$ is the partition 
in which $\overline{d+1}$, $1$, and $\overline{b+1}$ are three 
successive elements in a
block, and $a+1$, $\pm(mn-m+1)$, and $2$ are successive elements in 
another block. All other succession relations in $\pi$
are rotated by one unit in clockwise direction.
See \Cref{fig:13} for a schematic illustration of this operation.
\item If there is no block containing elements from the outer
{\it and} the inner circle, then let $a$,
$\overline{m(n-1)}$, and~$1$ be successive elements in a block.
{\em(}Again, the element~$a$ must be positive.{\em)}
By the ??artificial condition, the blocks on the inner circle
are $\{\overline{mn},mn-m+1,\dots,mn-1\}$ and its negative.
The image of $\pi$ under $\widetilde\phi$ is the partition 
in which $\overline{a+1}$, $1$, and $\bar2$ are successive
elements, and in which $\{\overline{mn-m+1},mn-m+2,\dots,mn-1,mn\}$
is a block. {\em(}This is the original block rotated by
{\em two} units in counter-clockwise direction{\em)}
See \Cref{fig:14} for a schematic illustration of this
operation.
\end{enumerate}
\end{proposition}

\begin{figure}
\begin{center}
  \begin{tikzpicture}[scale=1]

    \polygon{(-4,0)}{obj}{44}{2.5}
       {1,,,,a,,,,,,b,,,,e,,,d,,,,\hspace*{25pt}m(n-1),\overline{1},,,,\overline{a},,,,,,\overline{b},,,,\overline{e},,,\overline{d},,,,\overline{m(n-1)}\hspace*{25pt}}

    \polygoninner{(-4,0)}{objleftin}{8}{0.6}
      {,,,,,,,}

    % thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj1) to[bend right=50]
(obj5);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj23) to[bend right=50]
(obj27);

    % line connecting a and \overline{mn}
    \draw (obj44) to[bend right=50] (obj5);
    \draw (obj22) to[bend right=50] (obj27);

    % area for a+1 ... b-1
    \draw[dotted] (obj6) to[bend right=50] (obj10);
    \node at ($($0.92*($(obj8)+(4.0,0)$)$)-(4.0,0)$) {\tiny$X$};

    \draw[dotted] (obj28) to[bend right=50] (obj32);
    \node at ($($0.92*($(obj30)+(4.0,0)$)$)-(4.0,0)$) {\tiny$\overline{X}$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj40) to[bend right=20] (obj11) to[bend right=50] (obj15);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj15) to[bend left=10] (obj40);
    \draw (obj18) to[bend right=20] (obj33) to[bend right=50] (obj37);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj37) to[bend left=10] (obj18);

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{2.5}
       {1,2,,,,a+1,,,,,,b+1,,,,e+1,,,d+1,,,mn,\overline{1},\overline{2},,,,\overline{a+1},,,,,,\overline{b+1},,,,\overline{e+1},,,\overline{d+1},,,\overline{mn}}

    \polygoninner{(4,0)}{objleftin}{8}{0.6}
      {,,,,,,,}

    % thick line connecting 1 and \overline{d+1}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj41);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj19);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj2) to[bend right=50]
(obj6);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj24) to[bend right=50]
(obj28);

    % line connecting a+1 and b
    \draw (obj6) to[bend right=50] (obj12);
    \draw (obj28) to[bend right=50] (obj34);

    % line connecting b+1 and 2
    \draw (obj2) to[bend right=50] (obj12);
    \draw (obj24) to[bend right=50] (obj34);

    % area for a+2 ... b
    \draw[dotted] (obj7) to[bend right=50] (obj11);
    \node at ($($0.92*($(obj9)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj29) to[bend right=50] (obj33);
    \node at ($($0.92*($(obj31)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj1) to[bend right=20] (obj16);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj16) to[bend left=10] (obj41);
    \draw (obj23) to[bend right=20] (obj38);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 2pt on 2pt off 2pt on 2pt off 95pt on 2pt off 2pt on 2pt off
2pt on 2pt] (obj38) to[bend left=10] (obj19);

    \end{tikzpicture}
\end{center}
\caption{The action of the pseudo-rotation $\widetilde\phi$,
Case (1)}
\label{fig:12}
\end{figure}

\begin{figure}
\begin{center}
  \begin{tikzpicture}[scale=1]

    \polygon{(-4,0)}{obj}{44}{3}
       {1,,,,a,,,,,,b,,,,d,,,,,,,\hspace*{25pt}m(n-1),\overline{1},,,,\overline{a},,,,,,\overline{b},,,,\overline{d},,,,,,,\overline{m(n-1)}\hspace*{25pt}}

    \polygoninner{(-4,0)}{objin}{16}{1.2}
      {\pm mn\hspace*{5pt},,,,,,,,\hspace*{5pt}\mp mn,,,,,,}

    % thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj1) to[bend right=20]
(obj5);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj23) to[bend right=20]
(obj27);

    % line connecting a and \overline{mn}
    \draw (obj44) to[bend right=50] (obj5);
    \draw (obj22) to[bend right=50] (obj27);

    % area for d+1 ... m(n-1)-1
    \draw[dotted] (obj16) to[bend right=20] (obj21);
    \node at ($($0.92*($(obj19)+(4.0,0)$)$)-(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj38) to[bend right=20] (obj43);
    \node at ($($0.92*($(obj41)+(4.0,0)$)$)-(4.0,0)$) {\tiny$\overline{X}$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj11) to[bend right=30] (objin1) to[bend left=20] (obj15);
    \draw (obj33) to[bend right=30] (objin9) to[bend left=20] (obj37);

    % dotted line connecting b and d
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 25pt on 2pt off 2pt on 2pt off 2pt] (obj11) to[bend right=20]
(obj15);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 25pt on 2pt off 2pt on 2pt off 2pt] (obj33) to[bend right=20]
(obj37);

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{3}
       {1,2,,,,a+1,,,,,,b+1,,,,d+1,,,,,,\hspace*{25pt}m(n-1),\overline{1},\overline{2},,,,\overline{a+1},,,,,,\overline{b+1},,,,\overline{d+1},,,,,,\overline{m(n-1)}\hspace*{25pt}}

    \polygoninner{(4,0)}{objin}{16}{1.2}
      {,,\pm (m(n-1)+1)\hspace*{35pt},\mp
mn\hspace*{5pt},,,,,,,\hspace*{35pt}\mp (m(n-1)+1),\hspace*{5pt}\pm
mn,,,,}

    % thick line connecting 1 and \overline{mn}
%     \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
%     \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 2 and a+1
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj2) to[bend right=50]
(obj6);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj24) to[bend right=50]
(obj28);

    % line connecting a and \overline{mn}
    \draw (obj2) to[bend right=30] (objin3) to[bend left=20] (obj6);
    \draw (obj24) to[bend right=30] (objin11) to[bend left=20] (obj28);

    % area for d+1 ... m(n-1)-1
    \draw[dotted] (obj17) to[bend right=20] (obj22);
    \node at ($($0.92*($(obj20)-(4.0,0)$)$)+(4.0,0)$) {\tiny$X$};
    \draw[dotted] (obj39) to[bend right=20] (obj44);
    \node at ($($0.92*($(obj42)-(4.0,0)$)$)+(4.0,0)$) {\tiny$\overline{X}$};

    % line connecting \overlin{d} - b - e - ...
    \draw (obj12) to[bend right=20] (obj23);
    \draw (obj34) to[bend right=20] (obj1);
    \draw[line width=2.5pt,black] (obj23) to[bend left=20] (obj16);
    \draw[line width=2.5pt,black] (obj1) to[bend left=20] (obj38);

    % dotted line connecting b and d
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 25pt on 2pt off 2pt on 2pt off 2pt] (obj12) to[bend right=20]
(obj16);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 25pt on 2pt off 2pt on 2pt off 2pt] (obj34) to[bend right=20]
(obj38);

    \end{tikzpicture}
\end{center}
\caption{The action of the pseudo-rotation $\widetilde\phi$,
Case (2)}
\label{fig:13}
\end{figure}

\begin{figure}
\begin{center}
  \begin{tikzpicture}[scale=1]

    \polygon{(-4,0)}{obj}{44}{3}
       {1,,,,a,,,,,,,,,,,,,,,,,\hspace*{25pt}m(n-1),\overline{1},,,,\overline{a},,,,,,,,,,,,,,,,,\overline{m(n-1)}\hspace*{25pt}}

    \polygoninner{(-4,0)}{objin}{16}{1.2}
      {,,\overline{m(n-1)+1}\hspace*{25pt},mn,mn-1,,,,,,\hspace*{25pt}m(n-1)+1,\overline{mn},\overline{mn-1},,,}

    % thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend left=50] (obj44);
    \draw[line width=2.5pt,black] (obj23) to[bend left=50] (obj22);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj1) to[bend right=20]
(obj5);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj23) to[bend right=20]
(obj27);

    % line connecting a and \overline{mn}
    \draw (obj44) to[bend right=50] (obj5);
    \draw (obj22) to[bend right=50] (obj27);

    % line connecting \overlin{d} - b - e - ...
     \draw[dashed] (objin5) to[bend left=40] (objin6) to[bend left=40]
(objin7) to[bend left=40] (objin8) to[bend left=40] (objin9) to[bend
left=40] (objin10) to[bend left=40] (objin11);
     \draw (objin11) to[bend left=40] (objin12) to[bend
right=100,looseness=2] (objin5);
     \draw[dashed] (objin13) to[bend left=40] (objin14) to[bend
left=40] (objin15) to[bend left=40] (objin16) to[bend left=40]
(objin1) to[bend left=40] (objin2) to[bend left=40] (objin3);
     \draw (objin3) to[bend left=40] (objin4) to[bend
right=100,looseness=2] (objin13);

    \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygon{(4,0)}{obj}{44}{3}
       {1,2,,,,a+1,,,,,,,,,,,,,,,,\hspace*{25pt}m(n-1),\overline{1},\overline{2},,,,\overline{a+1},,,,,,,,,,,,,,,,\overline{m(n-1)}\hspace*{25pt}}

    \polygoninner{(4,0)}{objin}{16}{1.2}
      {,\overline{m(n-1)+2}\hspace*{25pt},\overline{m(n-1)+1}\hspace*{25pt},mn,,,,,,\hspace*{25pt}m(n-1)+2,\hspace*{25pt}m(n-1)+1,\overline{mn},,,,}

    % thick line connecting 1 and \overline{mn}
    \draw[line width=2.5pt,black] (obj1) to[bend right=40] (obj28);
    \draw[line width=2.5pt,black] (obj23) to[bend right=40] (obj6);

    % dotted line connecting 1 and a
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj2) to[bend right=10]
(obj6);
    \draw[dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on
2pt off 16pt on 2pt off 2pt on 2pt off 2pt] (obj24) to[bend right=10]
(obj28);

    % line connecting 2 and \overline{1}
    \draw (obj2) to[bend left=60] (obj23);
    \draw (obj24) to[bend left=60] (obj1);

    % line connecting \overlin{d} - b - e - ...
     \draw[dashed] (objin4) to[bend left=40] (objin5) to[bend left=40]
(objin6) to[bend left=40] (objin7) to[bend left=40] (objin8) to[bend
left=40] (objin9) to[bend left=40] (objin10);
     \draw (objin4) to[bend right=40] (objin3) to[bend
left=100,looseness=2] (objin10);
     \draw[dashed] (objin12) to[bend left=40] (objin13) to[bend
left=40] (objin14) to[bend left=40] (objin15) to[bend left=40]
(objin16) to[bend left=40] (objin1) to[bend left=40] (objin2);
     \draw (objin12) to[bend right=40] (objin11) to[bend
left=100,looseness=2] (objin2);


    \end{tikzpicture}
\end{center}
\caption{The action of the pseudo-rotation $\widetilde\phi$,
Case (3)}
\label{fig:14}
\end{figure}


\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygonnew{(-4,0)}{objleftout}{40}{3}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44}{1.1}
    \polygonnew{(-4,0)}{objleftin}{8}{0.75}
      {46,45,24,23,22,21,48,47}{0.65}

      
     \draw[fill=black,fill opacity=0.1] (objleftout40) to[bend right=30] (objleftout1) to[bend right=5] (objleftin1) to[bend right=50] (objleftin8) to[bend right=5] (objleftout40);

     \draw[fill=black,fill opacity=0.1] (objleftout2) to[bend right=30] (objleftout3) to[bend left=5] (objleftout16) to[bend right=10] (objleftin2) to[bend right=10] (objleftout2);

     \draw[fill=black,fill opacity=0.1] (objleftout4) to[bend right=30] (objleftout9) to[bend right=10] (objleftout10) to[bend right=30] (objleftout11) to[bend left=30] (objleftout4);

     \draw[fill=black,fill opacity=0.1] (objleftout5) to[bend right=30] (objleftout6) to[bend right=30] (objleftout7) to[bend right=30] (objleftout8) to[bend left=30] (objleftout5);

     \draw[fill=black,fill opacity=0.1] (objleftout12) to[bend right=30] (objleftout13) to[bend right=30] (objleftout14) to[bend right=30] (objleftout15) to[bend left=30] (objleftout12);

     \draw[fill=black,fill opacity=0.1] (objleftout17) to[bend right=30] (objleftout18) to[bend right=30] (objleftout19) to[bend right=5] (objleftin3) to[bend right=5] (objleftout17);

     \draw[fill=black,fill opacity=0.1] (objleftout20) to[bend right=30] (objleftout21) to[bend right=5] (objleftin5) to[bend right=50] (objleftin4) to[bend right=5] (objleftout20);

     \draw[fill=black,fill opacity=0.1] (objleftout22) to[bend right=30] (objleftout23) to[bend left=5] (objleftout36) to[bend right=10] (objleftin6) to[bend right=10] (objleftout22);

     \draw[fill=black,fill opacity=0.1] (objleftout24) to[bend right=30] (objleftout29) to[bend right=10] (objleftout30) to[bend right=30] (objleftout31) to[bend left=30] (objleftout24);

     \draw[fill=black,fill opacity=0.1] (objleftout25) to[bend right=30] (objleftout26) to[bend right=30] (objleftout27) to[bend right=30] (objleftout28) to[bend left=30] (objleftout25);

     \draw[fill=black,fill opacity=0.1] (objleftout32) to[bend right=30] (objleftout33) to[bend right=30] (objleftout34) to[bend right=30] (objleftout35) to[bend left=30] (objleftout32);

     \draw[fill=black,fill opacity=0.1] (objleftout37) to[bend right=30] (objleftout38) to[bend right=30] (objleftout39) to[bend right=5] (objleftin7) to[bend right=5] (objleftout37);

     \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygonnew{(4,0)}{objrightout}{40}{3}
      {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,1}{1.1}
    \polygonnew{(4,0)}{objrightin}{8}{0.75}
      {47,46,45,24,23,22,21,48}{0.65}

      
     \draw[fill=black,fill opacity=0.1] (objrightin7) to[bend right=10] (objrightout1) to[bend right=5] (objrightin1) to[bend right=50] (objrightin8) to[bend right=50] (objrightin7);

     \draw[fill=black,fill opacity=0.1] (objrightout2) to[bend right=30] (objrightout3) to[bend left=5] (objrightout16) to[bend right=10] (objrightin2) to[bend right=10] (objrightout2);

     \draw[fill=black,fill opacity=0.1] (objrightout4) to[bend right=30] (objrightout9) to[bend right=10] (objrightout10) to[bend right=30] (objrightout11) to[bend left=30] (objrightout4);

     \draw[fill=black,fill opacity=0.1] (objrightout5) to[bend right=30] (objrightout6) to[bend right=30] (objrightout7) to[bend right=30] (objrightout8) to[bend left=30] (objrightout5);

     \draw[fill=black,fill opacity=0.1] (objrightout12) to[bend right=30] (objrightout13) to[bend right=30] (objrightout14) to[bend right=30] (objrightout15) to[bend left=30] (objrightout12);

     \draw[fill=black,fill opacity=0.1] (objrightout17) to[bend right=30] (objrightout18) to[bend right=30] (objrightout19) to[bend right=30] (objrightout20) to[bend left=30] (objrightout17);

     \draw[fill=black,fill opacity=0.1] (objrightin3) to[bend right=10] (objrightout21) to[bend right=5] (objrightin5) to[bend right=50] (objrightin4) to[bend right=50] (objrightin3);

     \draw[fill=black,fill opacity=0.1] (objrightout22) to[bend right=30] (objrightout23) to[bend left=5] (objrightout36) to[bend right=10] (objrightin6) to[bend right=10] (objrightout22);

     \draw[fill=black,fill opacity=0.1] (objrightout24) to[bend right=30] (objrightout29) to[bend right=10] (objrightout30) to[bend right=30] (objrightout31) to[bend left=30] (objrightout24);

     \draw[fill=black,fill opacity=0.1] (objrightout25) to[bend right=30] (objrightout26) to[bend right=30] (objrightout27) to[bend right=30] (objrightout28) to[bend left=30] (objrightout25);

     \draw[fill=black,fill opacity=0.1] (objrightout32) to[bend right=30] (objrightout33) to[bend right=30] (objrightout34) to[bend right=30] (objrightout35) to[bend left=30] (objrightout32);

     \draw[fill=black,fill opacity=0.1] (objrightout37) to[bend right=30] (objrightout38) to[bend right=30] (objrightout39) to[bend right=30] (objrightout40) to[bend left=30] (objrightout37);
    \end{tikzpicture}
  \caption{The action of the map $\widetilde\phi$}
\label{fig:19}
\end{figure}
\begin{figure}
  \centering
  \begin{tikzpicture}[scale=1]
    \polygonnew{(-4,0)}{objleftout}{30}{3}
      {33,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,19,20,21,22,23,24,25,26,27,28,29,30,31,32}{1.1}
    \polygonnew{(-4,0)}{objleftin}{6}{0.75}
      {35,34,18,17,16,36}{0.65}

     \draw[fill=black,fill opacity=0.1] (objleftin6) to[bend right=5] (objleftout29) to[bend right=30] (objleftout30) to[bend right=5] (objleftin6);
     \draw[fill=black,fill opacity=0.1] (objleftin3) to[bend right=5] (objleftout14) to[bend right=30] (objleftout15) to[bend right=5] (objleftin3);

     \draw[fill=black,fill opacity=0.1] (objleftout2) to[bend right=5] (objleftin1) to[bend left=5] (objleftout1) to[bend right=10] (objleftout2);
     \draw[fill=black,fill opacity=0.1] (objleftout17) to[bend right=5] (objleftin4) to[bend left=5] (objleftout16) to[bend right=10] (objleftout17);

     \draw[fill=black,fill opacity=0.1] (objleftout3) to[bend right=30] (objleftout4) to[bend left=10] (objleftin2) to[bend right=5] (objleftout3);
     \draw[fill=black,fill opacity=0.1] (objleftout18) to[bend right=30] (objleftout19) to[bend left=10] (objleftin5) to[bend right=5] (objleftout18);

     \draw[fill=black,fill opacity=0.1] (objleftout5) to[bend right=30] (objleftout9) to[bend right=30] (objleftout13) to[bend left=5] (objleftout5);
     \draw[fill=black,fill opacity=0.1] (objleftout20) to[bend right=30] (objleftout24) to[bend right=30] (objleftout28) to[bend left=5] (objleftout20);

     \draw[fill=black,fill opacity=0.1] (objleftout6) to[bend right=30] (objleftout7) to[bend right=30] (objleftout8) to[bend left=30] (objleftout6);
     \draw[fill=black,fill opacity=0.1] (objleftout21) to[bend right=30] (objleftout22) to[bend right=30] (objleftout23) to[bend left=30] (objleftout21);

     \draw[fill=black,fill opacity=0.1] (objleftout10) to[bend right=30] (objleftout11) to[bend right=30] (objleftout12) to[bend left=30] (objleftout10);
     \draw[fill=black,fill opacity=0.1] (objleftout25) to[bend right=30] (objleftout26) to[bend right=30] (objleftout27) to[bend left=30] (objleftout25);

     \node[inner sep=0pt] at (0,0) {$\mapsto$};

    \polygonnew{(4,0)}{objrightout}{30}{3}
      {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33}{1.1}
    \polygonnew{(4,0)}{objrightin}{6}{0.75}
      {36,35,34,18,17,16}{0.65}

     \draw[fill=black,fill opacity=0.1] (objrightout1) to[bend left=30] (objrightout29) to[bend right=30] (objrightout30) to[bend right=30] (objrightout1);
     \draw[fill=black,fill opacity=0.1] (objrightout16) to[bend left=30] (objrightout14) to[bend right=30] (objrightout15) to[bend right=30] (objrightout16);

     \draw[fill=black,fill opacity=0.1] (objrightout2) to[bend right=5] (objrightin1) to[bend right=50] (objrightin6) to[bend right=10] (objrightout2);
     \draw[fill=black,fill opacity=0.1] (objrightout17) to[bend right=5] (objrightin4) to[bend right=50] (objrightin3) to[bend right=10] (objrightout17);

     \draw[fill=black,fill opacity=0.1] (objrightout3) to[bend right=30] (objrightout4) to[bend left=10] (objrightin2) to[bend right=5] (objrightout3);
     \draw[fill=black,fill opacity=0.1] (objrightout18) to[bend right=30] (objrightout19) to[bend left=10] (objrightin5) to[bend right=5] (objrightout18);

     \draw[fill=black,fill opacity=0.1] (objrightout5) to[bend right=30] (objrightout9) to[bend right=30] (objrightout13) to[bend left=5] (objrightout5);
     \draw[fill=black,fill opacity=0.1] (objrightout20) to[bend right=30] (objrightout24) to[bend right=30] (objrightout28) to[bend left=5] (objrightout20);

     \draw[fill=black,fill opacity=0.1] (objrightout6) to[bend right=30] (objrightout7) to[bend right=30] (objrightout8) to[bend left=30] (objrightout6);
     \draw[fill=black,fill opacity=0.1] (objrightout21) to[bend right=30] (objrightout22) to[bend right=30] (objrightout23) to[bend left=30] (objrightout21);

     \draw[fill=black,fill opacity=0.1] (objrightout10) to[bend right=30] (objrightout11) to[bend right=30] (objrightout12) to[bend left=30] (objrightout10);
     \draw[fill=black,fill opacity=0.1] (objrightout25) to[bend right=30] (objrightout26) to[bend right=30] (objrightout27) to[bend left=30] (objrightout25);
    \end{tikzpicture}
  \caption{The action of the map $\widetilde\phi$}
\label{fig:20}
\end{figure}

\begin{remark}
??\Cref{fig:17} provides a schematic illustration of 
the construction in the statement of 
the above proposition, while \Cref{fig:18} provides a concrete
example. In this example, $m=3$, $n=8$, $a=14$,
$b_1=15$, and $b_2=19$.
\end{remark}

\begin{proof}[Proof of \Cref{prop:2D}]
Let us first assume that, 
in the $m$-divisible type-$D$ non-crossing partition~$\pi$,
the block of $\overline{m(n-1)}$ contains another negative element
which is different from~$\overline{mn-1}$.
Among those, let $\overline{mi_1-1}$ be the one which is the
predecessor of $\overline{m(n-1)}$ in the block.
Obviously, we have $i_1>0$.
\Cref{fig:20} provides a sketch of this situation. 
Let $(w_0,w_1,\dots,w_m)$ denote the
element of $NC^{(m)}_+(D_n)$ which corresponds to $\pi$ under the
bijection $\Na^m_{D_n}$. By the definition of $\Na^m_{D_n}$, we
see that $w_{m-1}(n-1)=i_1$. From
\Cref{lem:LRD}.(1), we infer $w_{m-1}^L=w_{m-1}$ and
$w_{m-1}^R=\one$. This means that the pseudo-rotation \eqref{eq:phiD}
reduces to ordinary rotation in this special case, as we claimed. 

\begin{figure}
??
  \caption{The elements $\overline{m(n-1)}$ and $1$ are not in the same block}
\label{fig:21}
\end{figure}

\medskip
For the remainder of the proof, we assume that the block of
$\overline{m(n-1)}$ contains no other negative element which
is different from~$\overline{mn-1}$. By arguments that are completely
analogous to the corresponding ones in the proof of
\Cref{prop:2B}, one shows that~$1$ is an element 
of that block.

\medskip
Let again $(w_0,w_1,\dots,w_m)$ denote the
element of $NC^{(m)}_+(D_n)$ which corresponds to $\pi$ under the
bijection $\Na^m_{D_n}$. 

\smallskip
{\sc Case (1).} 
This is proved in the same way as Case~(1) in the proof of
\Cref{prop:2B}.

\smallskip
{\sc Case (2).} This case is also similar to Case~(1) in the proof
of \Cref{prop:2B}. We are at times sketchy here,
leaving some details to the reader.

By the assumptions of this case, and by what we already
showed, we know that $a$, $\overline{m(n-1)}$, and~$1$
are successive elements in a block, and that
$\bar d$, $\mp{mn}$ and $\bar b$ are successive elements
in a block.

We write $a=mA-1$, $b=mB+1$, and $d=mD-1$,
for suitable integers~$A$, $B$, and $D$. Here, $D$ is positive.
The integer $A$ can be between $1$ and $n-1$ or equal to $\mp n$,
and $B$ can be between $A$ and $D-1$ or equal to $\mp(n-1)$.
In view of the above observations,
by the definition of  $\Na^m_{D_n}$, we have
\begin{align}
\notag
%\label{eq:wmex1}
w_{m-1}(n-1)&=A,\\
\notag
%\label{eq:wmex1}
w_{m-1}(\pm n)&=D,\\
\label{eq:wmex4c}
w_{m}(n-1)&=n-1,\\
\label{eq:wmex4d}
w_m(B)&=\pm n.
\end{align}
Consequently, the cycle of $w_{m-1}$ containing $n-1$ is of the
form 
$$((D,\dots,n-1,\bar A,\dots,\pm n)).$$
(At this point, one has to argue that $n-1$ and $\pm n$ are indeed
in the same cycle of $w_{m-1}$.)
By \Cref{lem:LRD}.(3), we have 
$$w^R_{m-1}=((n-1,\pm n)),\quad \text{and}\quad 
w^L_{m-1}\text{ contains the cycles }((D,\dots,n-1))((\bar A,\dots,n)).$$
Together with \eqref{eq:wmex4c} and \eqref{eq:wmex4d}, 
this implies the relations
\begin{align*}
%\label{eq:wmex1}
w^L_{m-1}(n-1)&=D,\\
%\label{eq:wmex1}
w^L_{m-1}(n)&=\bar A,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(1)&=\pm n,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(B+1)&=\bar 1.
\end{align*}
By the definition of $\Na^m_{D_n}$, this means that
$\overline{d+1}$, $1$, and~$\overline{b+1}$ are successive elements in 
their block, that
$a+1$, $\pm(mn-m+1)$, and~$2$ are successive elements in 
their block, 
while all other ``block connections" are rotated by one unit
 in clockwise direction (meaning clockwise on the outer circle
and counter-clockwise on the inner circle).
This is in accordance with the corresponding assertion in the
proposition.

\smallskip
{\sc Case (3).} 
We are again somewhat sketchy here.
We write again $a=mA-1$ for a suitable positive integer~$A$.
By the definition of  $\Na^m_{D_n}$,
we have
\begin{align}
\notag
%\label{eq:wmex1}
w_{m-1}(n-1)&=\bar A,\\
\notag
%\label{eq:wmex1}
w_{m-1}(n)&=\bar n,\\
\label{eq:wmex4f}
w_m(n-1)&=n-1,\\
\label{eq:wmex4g}
w_m(n)&=n.
\end{align}
Consequently, the cycle of $w_{m-1}$ containing $n-1$ is of the
form 
$$[A,\dots,\dots,n-1][n].$$
By \Cref{lem:LRD}.(4), we have 
$$w^R_{m-1}=[n-1][n],\quad \text{and }\quad 
w^L_{m-1}\text{ contains the cycles }((A,\dots,n-1))((n)).$$
Together with \eqref{eq:wmex4f} and \eqref{eq:wmex4g}, 
this implies the relations
\begin{align*}
%\label{eq:wmex1}
w^L_{m-1}(n-1)&=A,\\
%\label{eq:wmex1}
w^L_{m-1}(n)&=n,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(1)&=\bar 1,\\
%\label{eq:wmex3}
(cw^R_{m-1}w_mc^{-1})(n)&=\bar n.
\end{align*}
By the definition of $\Na^m_{D_n}$, this means that
$\overline{a+1}$, $1$, and~$\bar 2$ are successive elements in 
their block, that $mn$, $\overline{mn-m+1}$, and $mn-m+2$
are successive elements in their block,
while all other ``block connections" are rotated by one unit
 in clockwise direction (meaning clockwise on the outer circle
and counter-clockwise on the inner circle).
This is in accordance with the corresponding assertion in the
proposition.

This completes the proof of the proposition.
\end{proof}



\subsection{Enumeration}
\label{sec:enumD}

\subsection{Pseudo-rotationally invariant elements and cyclic sieving}
\label{sec:rotD}

\subsubsection*{Cyclic sieving}

\section{Positive \texorpdfstring{$m$}{m}-divisible non-crossing partitions in dihedral and exceptional types}
\label{sec:typeExc}

In this section, we study the two variants of the positive
Kreweras complements in the
dihedral and exceptional types. 
Before discussing each type individually, we collect some
considerations that uniformly hold for all types.
%CK: Zweimal hintereinander "This"
%This
Our approach uses \Cref{cor:poskrewwords} to derive versions of \Cref{cor:positivefusscatcounting} counting subwords that are symmetric under powers of the positive Kreweras complements.
This provides a recipe for a given finite Coxeter system $(W,\reflS)$ how to count all $m$-divisible non-crossing partitions that are invariant under powers of the 
%CK: positie
positive Kreweras complements \emph{for all}~$m$ 
%CK: "together" ergibt keinen Sinn hier. 
%together 
using a finite computation.

\medskip

First, we have seen that any two Coxeter elements~$c$ and~$c'$ are conjugate, and that this conjugation induces a bijection between $\mNCplus$ and ${N\!C^{(m)}_+(W,c')}$.
It moreover follows from the definition that this bijection is $\Krewplus$-equivariant.
It therefore suffices to consider the bipartite Coxeter element~$c = c_A c_B$
given by the decomposition $\reflS = A \sqcup B$ such that all simple generators in~$A$ and in~$B$ pairwise commute, and where we write $c_A$ and $c_B$ for the
products of the generators in~$A$ and in~$B$, respectively. 
This has the computational advantage that we have
\begin{equation*}
  \Rword(\wo) =  \underbrace{c_A\ c_B\ c_A\ \cdots}_h.
\end{equation*}
%CK: This is,
That is, $\Rword(\wo)$ consists of~$h$ alternating copies of $c_A$ and $c_B$.
We thus do not need to consider the situation ``up to commutations'' as there are no commutations 
%CK: "happening" kommt mir komisch vor.
%happening
occurring in this Coxeter-sorting word.
As $\invc^m\invc^L$ then consists of $m+1$ concatenated copies of $\invc$ with the last~$n$ letters removed, we identify the word with $(m+1)h-2$ alternating copies of $c_A$ and $c_B$.
More concretely, we distinguish letters in the word $\invc^m\invc^L$ by identifying their positions using the word in the alternating copies of $c_A$ and $c_B$, while we take the product directly inside $\invc^m\invc^L$.
Similarly, $\invc^{m-1}\invc^L$ then involves $mh-2$ alternating copies of $c_A$ and $c_B$.
This additional regularity (that is in general missing for arbitrary Coxeter elements) makes the description of subwords that are invariant under powers of the positive Kreweras complements easier to describe.

\medskip

We next 
%CK: "determine" ist eigenartig in diesem Zusammenhang.
%determine
describe how to find the appropriate numerical parameters for the situation that one considers an $\ell$-fold symmetry.
To this end, we make use of the following lemma.

\begin{lemma}\label{lem:numbercrunching}
  Let $A,B,C$ be natural numbers without a common factor, and let~$\mathcal{Y}$ be the set of all integers~$Y$ such that~$A$ divides $BY+C$.
  Then
  \begin{enumerate}
    \item $\mathcal{Y} = \emptyset$ if~$A$ and~$B$ have a common factor, and \label{eq:nosolution}
    \item $\mathcal{Y} = \{ Y : Y \equiv Y_0~\text{\em (mod }A)\}$ for a fixed $Y_0$ if~$A$ and~$B$ do not have a common factor.
  \end{enumerate}
\end{lemma}
\begin{proof}
  Let~$Y$ be an integer such that~$A$ divides $BY+C$.
  Phrased differently, $BY+C \equiv 0~\text{(mod }A)$.
  If~$A$ and~$B$ have a common factor (which is necessarily not a factor of~$C$), then this equation does not have a solution.
  Otherwise,~$B$ has a multiplicative inverse $B'$ modulo~$A$, meaning there exists an integer $B'$ such that $B'B \equiv 1~\text{(mod }A)$.
  This implies $Y \equiv -B'C~\text{(mod }A)$, as desired.
\end{proof}

One now applies this lemma to the situation 
%CK: that
where~$\ell$ divides $(m+1)h-2$.
First, we set $y = 2$ if~$\ell$ and~$h$ are even, and $y=1$ otherwise.
\Cref{lem:numbercrunching} then says that there exist an~$m$ such that~$\ell$ divides $(m+1)h-2$ if and only if~$\ell/y$ and~$h/y$ do not have a common factor.
In this case,~$m$ is given by the fixed 
%CK: modifiziert
residue class~$m'$ modulo $\ell/y$, $0 \leq m' < \ell/y$, given by
\begin{equation}
  m \equiv m' = (h/y)^{-1}(2-h) \mod{\ell/y}.
\end{equation}
\cs{recheck}
In this case, we moreover obtain
\begin{equation}
  \frac{y}{\ell}\Big((m+1)h-2\Big) = ah + \frac{y}{\ell}\Big((m'+1)h-2\Big)
\end{equation}
for the parameter~$a$ given by $m = a\frac{\ell}{y} + m'$, and integral $\frac{y}{\ell}\big((m'+1)h-2\big)$.
We thus define in this case the reflection~$\r$ inside $\invc$ to be the letter in position
\[
  \ifix = \frac{n}{2}\frac{y}{\ell}\big((m'+1)h-2\big)+1.
\]
\cs{explain the over 2 in $\frac{n}{2}$}

\medskip

We 
%CK: Was soll "mainly" hier bedeuten? Das suggeriert, dasz der andere
%    Fall von geringerem INteresse waere. Das trifft aber nicht zu.
%    Er musz ebenfalls abgehandelt werden.
mainly consider the case $\psi \equiv \one$.
Under this condition, 
%CK: we have that 
the Coxeter number~$h$ is even.
Let~$\zeta$ be a primitive $\big((m+1)h-2\big)$\th\ root of unity.
According to \Cref{thm:order}, this corresponds to twice the order of $\Krewplus$ and of $\Krewplustilde$ 
%CK: Worauf bezieht sich "in this case"?
%    Mir kommt vor, dasz man das ersatzlos streichen kann.
in this case.

\medskip

The case $\psi \not\equiv \one$ is indeed slightly harder to approach 
%CK: as
since the key observation in \Cref{lem:invariantsitting} does not hold in this simple form.
We will briefly sketch in \Cref{rem:E6} how to handle the case $\psi\not\equiv\one$ in general, but we will not do it in detail, as it turns out that we have to use that alternative description only in type~$E_6$ where one can as well use the argument in an ad hoc manner.

\medskip

We have to show that $\mCatplus(W;q)$ and $\mCatplus[m+1](W;q)$, evaluated at $\zeta^{2k}$ (which is a primitive $(\ell/2)$\th\ root of unity), equals the number of positive $m$-divisible non-crossing partitions that are invariant under $(\Krewplus)^{k}$ and $(\Krewplustilde)^k$, respectively.
Observe that \Cref{cor:poskrewwords} implies that this corresponds to
an $\ell/2$-fold symmetry of the letters inside $\invc^m\invc^R$.
Moreover, this $(\ell/2)$-fold symmetry needs to preserve the considered subwords.
The orbit of a letter inside $\invc^m\invc^L$ under $\Krewplus$ exactly corresponds to the $(m+1)h/2-1$ copies of this letter inside the $(m+1)h/2-1$ copies of $c = c_A c_B$.
We thus obtain the following proposition.

\begin{lemma}\label{lem:invariantsitting}
  Let~$\r_1,\dots,\r_{\lenR(w)}$ be a subword of $\invc^m\invc^L$ such that $w = r_1\cdots r_{\lenR(w)} \in \NCplus$.
  Then this subword is invariant under the 
%CK: $k$-times
$k$-fold application of $\Krewplustilde$ if and only it is invariant under the cyclic shift sending a letter in the $j$\th\ copy of $c = c_Ac_B$ inside $\invc^m\invc^L$ to the same letter in the $(j+k)$\th\ copy of~$c$.
\end{lemma}
\begin{proof}
  This follows from \Cref{cor:poskrewwords} as discussed in the paragraph before the lemma.
\end{proof}

This lemma has the immediate consequence that we can exactly determine the orbits of the letters.
\begin{corollary}\label{cor:invariantshift}
  In the situation of the previous lemma, 
%CK: "we have that" sollte man moeglichst vermeiden.
the integer $\ell/2$ divides $\lenR(w)$.
%CK: Umformuliert.
With $k'$ being defined by $\lenR(w) = \ell/2 \cdot k'$), 
the orbits of the letters $\r_1,\dots,\r_{\lenR(w)}$ are those letters that are~$k'$ apart, giving $k'$ orbits each of size~$\ell/2$.
\end{corollary}

This corollary implies the first important 
%CK: restiction
restriction, as a priori one has to consider an infinite number of parameters~$\ell/2$, while now we only need to consider divisors of $n' = \lenR(w) \leq n$ for all different $w \in \NCplus$.
%CK: This is,
In other words, 
%CK: Die \emph-Betonung hier sieht inkonsequent aus. Entweder gehoert
%    "such that" noch dazu, oder sie mueszte - sagen wir - auf "only"
%    beschraenken.
\emph{we only need to consider all}~$\ell$ such that $1 \leq \ell/2 \leq n$.

%CK: Das "It" bezieht sich worauf? Das musz umformuliert werden.
%    "written as a computationally tractable way" klingt auch sehr komisch.
It can moreover be written as a computationally tractable way to obtain a necessary and sufficient condition on a reduced factorisation $\r_1\cdots\r_{\lenR(w)}$ of a non-crossing partition $w = r_1\cdots r_{\lenR{w}} \in \NC$ to correspond to an invariant $m$-divisible non-crossing partition for large enough~$m$.
We will then later explicitly count the number of subwords of $\invc^m\invc^L$ that are invariant under the positive Kreweras complement 
%CK: Das meinst Du wohl nicht? word -> number ??
that spell out this word as a polynomial in the parameter~$m$.

\begin{proposition}\label{prop:invariantsittingdetails}
  Let $c = r_1\cdots r_{n'}$ be a reduced $\reflR$-word for $w \in \NC$, and let $i_1,\ldots,i_{n'}$ be the sequence of indices of the reflections  $r_1,\ldots,r_{n'}$ inside $\invc$.
  Then there exists an $(\ell/2)$-fold invariant subword of $\invc^m\invc^L$ for some~$m$ 
%CK: Ich nehme an, dasz mit "this word" das "reduced $\reflR$-word"
%    im ersten Satz gemeint ist. Das musz man aber erraten. 
%    Im ersten Satz sollte dies einen Namen bekommen, auf den dann
%    Bezug genommen wird.
spelling out this word if and only if $\ell/2$ divides~$n'$ and,
with $k'$ defined by $n' = \ell/2 \cdot k'$, the cyclic distances
  \[
    a_1 = i_{k'+1}-i_1, a_2 = i_{k'+2}-i_2,\ldots, a_{n'-1} = i_{k'-1}-i_{n'-1}, a_{n'} = i_{k'}-i_{n'} \pmod{N}
  \]
  inside $\invc$ satisfy
  \begin{enumerate}
    \item $a_1 \equiv \cdots \equiv a_{n'} \equiv 0 \pmod{kn}$, and
    \item $a_1 = a_2 = \dots = a_{n'-k'} = a_{n'-k'+1} + n = \dots a_{n'} + n$.
  \end{enumerate}
\end{proposition}
\begin{proof}
  First assume that there is an $(\ell/2)$-fold invariant subword of $\invc^m\invc^L$ spelling out $r_1\dots r_{n'}$.
  This means that this subword is invariant under the $k$-fold application of $\Krewplustilde$.

We have seen in \Cref{cor:invariantshift} that orbits of the reflections $r_1,\dots,r_{n'}$ in this word under $(\Krewplustilde)^k$ must be all letters that are~$k'$ apart.
  This is exactly the first numerical condition.
  Moreover, the letters that are~$k'$ apart must be spread equally inside the word $\invc^m\invc^L$, implying the second numerical condition, where the shift by~$n$ in $a_{n'-k'+1},\ldots,a_{n'}$ 
%CK: come
comes from the fact that these distances have passed the end of the word $\invc^m\invc^L$ and there are exactly~$n$ letters missing in the final $\invc^L$.

  \smallskip

  On the other hand, if these numerical conditions are satisfied for a reduced $\reflR$-word $\r_1\cdots\r_{n'}$ of some $w \in \NC$, then one can always produce an~$m$ such that $\r_1\cdots\r_{n'}$ fits properly into~$\invc^m\invc^L$ as described in \Cref{lem:invariantsitting}.
\end{proof}

Given $\ell/2$, \Cref{prop:invariantsittingdetails} provides an algorithm to determine which $w \in \NC$ and which reduced $\reflR$-words of~$w$ possibly contribute to the counting formula of $\ell/2$-fold symmetric $m$-divisible non-crossing partitions.
It is now only left to determine which~$m$ they contribute to and how much they actually contribute.

\medskip

%CK:This
The letter~$\r$ is 
%CK: Soll es das meinen?
%specified to satisfy
characterised by the following lemma.

\begin{lemma} \label{lem:missingletter}
  Let the word $\invc^m\invc^L$ have an $\ell/2$-fold symmetry.
  It then consists of $\ell/2$ copies of $\invc^a$ followed by the fraction of $\invc$ where the first missing reflection is~$\r$.
  In particular, the fundamental domain for the $\ell/2$-fold symmetry of the word $\invc^m\invc^L$ is given by a numerical computation which is independent of the parameter~$m$.
\end{lemma}
\begin{proof}
  This was exactly the way the reflection~$\r$ was constructed.
\end{proof}

We are finally in the position to compute the number of $m$-divisible non-crossing partitions  that are invariant under 
%CK: Soll das "and" statt "und" heiszen, oder ist da mehr schiefgegangen?
the $\ell/2$-fold symmetry und $\Krewplus$ 
and the number of $(m+1)$-divisible non-crossing partitions that are invariant under 
%CK: Soll das "and" statt "und" heiszen, oder ist da mehr schiefgegangen?
the $\ell/2$-fold symmetry und $\Krewplustilde$.

\begin{corollary}\label{cor:countingbinomial}
  Let $a = \lfloor m/\frac{\ell}{y} \rfloor$.
  Then the sum
  \begin{equation}
    \sum \binom{a+\asc(i_1,\ldots,i_{k'},\ifix)}{k'},
  \end{equation}
  counts
  \begin{enumerate}
    \item positive $m$-divisible non-crossing partitions $\mNCplus$ that are invariant under the $k$-fold application of $\Krewplus$ if the sum ranges over all factorisations $c=r_1\cdots r_n$ of the Coxeter element satisfying the two properties in \Cref{prop:invariantsittingdetails}, and where~$k'$ is given by $n = \ell/2 \cdot k'$, and
    \item positive $(m+1)$-divisible non-crossing partitions $\mNCplus[m+1]$ that are invariant under the $k$-fold application of $\Krewplustilde$ if the sum ranges over all factorisations $r_1\cdots r_{n'}$ of non-crossing partitions $\NC$ satisfying the two properties in \Cref{prop:invariantsittingdetails}, 
%CK: Warum schreibst Du hier "depends on the factorisation" und vorher nicht?
%    Und was bedeutet das eigentlich? 
and where~$k'$ depends on the factorisation and is given by $n' = \ell/2 \cdot k'$.
  \end{enumerate}
  In both cases, $i_1,\ldots,i_{k'}$ is the sequence of indices of the reflections  $r_1,\ldots,r_{k'}$ inside $\invc$.
\end{corollary}
\begin{proof}
  Given the previous considerations about the fundamental domain for the $\ell/2$-fold symmetry, the proof is the same as the proof of \Cref{cor:positivefusscatcounting}.
  In particular, the formula given here becomes 
%CK: Soll es das heiszen?
%that counting formula
the counting formula of that corollary in the case of $\ell = 2$.
\end{proof}

To finish these considerations, we provide the steps to count $m$-divisible non-crossing partitions that are invariant under an $\ell/2$-fold symmetry, and compare it to the evaluation of the $q$-extension of the positive Fu\ss-Catalan numbers at the corresponding primitive root of unity:
\begin{enumerate}
  \item Compute all reduced $\reflR$-words of the Coxeter element $c = c_A c_B$ using the transitivity of the Hurwitz action.\cs{add reference}
  \begin{enumerate}
    \item If one wants to compute the symmetry under $\Krewplustilde$, one then computes as well all prefixes of these reduced $\reflR$-words.
  \end{enumerate}
  \item Represent all reduced words for~$c$ or for all of $\NC$ as indices of subwords of $\invc$.
  \item Use \Cref{prop:invariantsittingdetails} to determine which factorisations contribute to the enumeration. Disregard all factorisations not satisfying these properties.
  \item Use \Cref{cor:countingbinomial} to determine the contribution of each factorisation as a polynomial in the parameter~$a$.
  \item Evaluate $\mCatplus(W;q)$ or $\mCatplus[m+1](W;q)$ (depending on 
whether we consider $\Krewplus$ or $\Krewplustilde$, respectively) at the primitive root~$\zeta^{2k}$ and express the result in terms of the parameter~$a$.
\end{enumerate}

Before using these observations for the dihedral and the exceptional types, we have to discuss the case when $\psi\not\equiv\one$.
Observe that this only applies to the dihedral types $I_2(h)$ with odd~$h$ and to type~$E_6$.
The difference is that after the positive Kreweras complement is applied $(m+1)h-2$ times 
%CK: Hoert hier der getippte Text auf, oder ist da etwas passiert?


\subsection{Cyclic sieving: the dihedral groups}
\label{sec:siev0}

\cs{tba}

\subsection{Cyclic sieving for $\Krewplus$: the exceptional groups}
\label{sec:siev2}

In each case, we will briefly determine the parameters in the above computation.
These are then used in the algorithm implemented in the computer algebra system SageMath\footnote{The computations were performed using \texttt{sage-7.2}. The development was supported by the project ``Combinatorial and geometric structures for reflection groups and groupoids'' within the German Research Foundation priority program ``Algorithmic and experimental methods in Algebra, Geometry, and Number Theory''. A worksheet with the code and examples is available upon request.}.

\subsubsection*{\sc\bf Case $H_3$}
The degrees are $2,6,10$, and $(m+1)h-2 = 10m+8 = k \cdot \ell$. We hence have
\begin{align}
\mCatplus(H_3;q)=\frac 
{[10m+8]_q\, [10m+4]_q\, [10m]_q} 
{[10]_q\, [6]_q\, [2]_q} .\label{eq2:H3}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity. 
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:H3.1}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=\mCatplus(H_3),
&\text{if }\ell \in\{ 1,2\},\\
\label{eq2:H3.2}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=\tfrac {5m+4}3,
&\text{if }m\equiv1\text{ (mod $3$)},\ \ell \in \{ 3,6\},\\
\label{eq2:H3.3}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

We may assume that $\ell$ is even.
In this case $\ell/2$ divides~$n=3$, implying $\ell \in \{2,6\}$.

The case $\ell = 2$ is computed using all ascent-descent configurations of the $50$ factorisations of~$c$.

The case $\ell = 6$ is given by the parameters
\begin{align*}
  y = 2,\quad m' = 1,\quad \ifix = 9.
\end{align*}
There are $5$ factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}.
As positions of letters in $\invc$ (which lists the~$15$ reflections in the reflection ordering), these are given by
\[
  \big\{(2, 11, 5), (5, 14, 8), (8, 2, 11), (11, 5, 14), (14, 8, 2) \big\}.
\]
We thus compute the number of $(\Krewplus)^k$-invariant subwords as
\[
  3\binom{a+1}{1} + 2\binom{a}{1} = 5a+3,
\]
where the first $3$ come from the $3$ factorisations with ascents $i_1 < 9$, while the second $2$ come from the $2$ factorisations with descents $i_1 > 9$.
With $m' = 1$, we obtain $m = 3a+1$, which, by \Cref{eq2:H3.2}, leads to
\[
  \frac{5m+4}{3} = \frac{15a+9}{3} = 5a+3,
\]
as desired.

\subsubsection*{\sc\bf Case $H_4$}
The degrees are $2,12,20,30$, and $(m+1)h-2 = 30m+28 = k \cdot \ell$. We hence have
\begin{align}
\mCatplus(H_4;q)=\frac 
{[30m+28]_q\, [30m+18]_q\, [30m+10]_q\, [30m]_q} 
{[30]_q\, [20]_q\, [12]_q\, [2]_q} .\label{eq2:H4}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity. 
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:H4.1}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=\mCatplus(H_4),
&\text{if }\ell \in \{1,2\},\\
\label{eq2:H4.2}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=\tfrac {(15m+14)m}4,
&\text{if } m\equiv0\text{ (mod $2$)},\ \ell = 4,\\
\label{eq2:H4.3}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

We may assume that $\ell$ is even.
In this case $\ell/2$ divides~$n=4$, implying $\ell \in \{2,4,8\}$.

The case $\ell = 2$ is computed using all ascent-descent configurations of the $1150$ factorisations of~$c$.

The case $\ell = 4$ is given by the parameters
\begin{align*}
  y = 2,\quad m' = 0,\quad \ifix =  29.
\end{align*}
There are $30$ factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}.
As positions of letters in $\invc$ (which lists the~$60$ reflections in the reflection ordering), these are given by
\[
  \left\{\begin{array}{ccccc}
  (4, 44, 32, 12), & (4, 52, 32, 20), & (8, 48, 36, 16), & (8, 56, 36, 24), & (12, 52, 40, 20), \\
  (12, 60, 40, 28), & (16, 4, 44, 32), & (16, 56, 44, 24), & (20, 8, 48, 36), & (20, 60, 48, 28), \\
  (24, 4, 52, 32), & (24, 12, 52, 40), & (28, 8, 56, 36), & (28, 16, 56, 44), & (32, 12, 60, 40), \\
  (32, 20, 60, 48), & (36, 16, 4, 44), & (36, 24, 4, 52), & (40, 20, 8, 48), & (40, 28, 8, 56), \\
  (44, 24, 12, 52), & (44, 32, 12, 60), & (48, 28, 16, 56), & (48, 36, 16, 4), & (52, 32, 20, 60),\\
  (52, 40, 20, 8), & (56, 36, 24, 4), & (56, 44, 24, 12), & (60, 40, 28, 8), & (60, 48, 28, 16)
  \end{array}
  \right\}
\]
We thus compute the number of $(\Krewplus)^k$-invariant subwords as
\[
  22\binom{a+1}{2} + 8\binom{a}{2} = (15a+7)a,
\]
where the first $22$ come from $8$ situations of the form $i_1 < i_2 > 29$ and $14$ situations of the form $i_1 > i_2 < 29$, while the second~$8$ come from~$8$ situations of the form $i_1 > i_2 > 29$.
With $m' = 0$, we obtain $m = 2a$, which, by \Cref{eq2:H4.2}, leads to
\[
  \frac{(15m+14)m}{4} = \frac{(30a+14)2a}{4} = (15a+7)a,
\]
as desired.

The case $\ell = 8$ is given by the parameters
\[
  y = 2,\quad m' = 2,\quad \ifix =  45.
\]
In this case, there are no factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}, in agreement with \Cref{eq2:H4.3}.

\subsubsection*{\sc\bf Case $F_4$}
The degrees are $2,6,8,12$, and $(m+1)h-2 = 12m+10 = k \cdot \ell$. We hence have
\begin{align}
\mCatplus(F_4;q)=\frac 
{[12m+10]_q\, [12m+6]_q\, [12m+4]_q\, [12m]_q} 
{[12]_q\, [8]_q\, [6]_q\, [2]_q} .\label{eq2:F4}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:F4.1}
\lim_{q\to\zeta}\mCatplus(F_4;q)&=\mCatplus(F_4),
&\text{if }\ell \in \{1,2\},\\
\label{eq2:F4.2}
\lim_{q\to\zeta}\mCatplus(F_4;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

We may assume that $\ell$ is even.
In this case $\ell/2$ divides~$n=4$, implying $\ell \in \{2,4,8\}$.

The case $\ell = 2$ is computed using all ascent-descent configurations of the $432$ factorisations of~$c$.

The cases $\ell \in \{4,8\}$ reduce to \Cref{lem:numbercrunching}\eqref{eq:nosolution} as we have to consider that $\ell$ divides $12m+10$ in this case.

\subsubsection*{\sc\bf Case $E_6$}
The degrees are $2,5,6,8,9,12$, and $(m+1)h-2 = 12m+10 = k \cdot \ell$. We hence have
\begin{align}
\mCatplus(E_6;q)=\frac 
{[12m+10]_q\, [12m+7]_q\, [12m+6]_q\, [12m+4]_q\, [12m+3]_q\, [12m]_q} 
{[12]_q\, [9]_q\, [8]_q\, [6]_q\, [5]_q\, [2]_q} .\label{eq2:E6}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:E6.1}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=\mCatplus(E_6),
&\text{if }\ell=1,\\
\label{eq2:E6.2}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=\tfrac {(6m+5)(2m+1)(3m+1)m}2,
&\text{if }\ell= 2,\\
%\label{eq2:E6.3}
%\lim_{q\to\zeta}\mCatplus(E_6;q)&=\tfrac {12m+10}5,
%&\text{if }\ell=5,\ m\equiv0\text{ (mod $5$)},\\
\label{eq2:E6.4}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

\cs{tba}

\subsubsection*{\sc\bf Case $E_7$}
The degrees are $2,6,8,10,12,14,18$, and $(m+1)h-2 = 18m+16 = k \cdot \ell$. We hence have
\begin{multline}
\mCatplus(E_7;q)=\frac 
{[18m+16]_q\, [18m+12]_q\, [18m+10]_q} 
{[18]_q\, [14]_q\, [12]_q}\\
\times
\frac 
{[18m+8]_q\, [18m+6]_q\, [18m+4]_q\, [18m]_q} 
{[10]_q\, [8]_q\, [6]_q\, [2]_q} .\label{eq2:E7}
\end{multline}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:E7.1}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\mCatplus(E_7),
&\text{if }\ell \in \{ 1,2 \},\\
\label{eq2:E7.2}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\tfrac {9m+8}7,
&\text{if }  m\equiv3\text{ (mod $7$)},\ \ell\in \{7,14\},\\
\label{eq2:E7.3}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

We may assume that $\ell$ is even.
In this case $\ell/2$ divides~$n=7$, implying $\ell \in \{2,14\}$.

The case $\ell = 2$ is computed using all ascent-descent configurations of the $1{,}062{,}882$ factorisations of~$c$.

The case $\ell = 14$ is given by the parameters
\begin{align*}
  y = 2,\quad m' = 3,\quad \ifix =  36.
\end{align*}
There are $9$ factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}.
As positions of letters in $\invc$ (which lists the~$63$ reflections in the reflection ordering), these are given by
\[
  \left\{\begin{array}{ccc}
 (7, 42, 14, 49, 21, 56, 28), &
 (14, 49, 21, 56, 28, 63, 35),&
 (21, 56, 28, 63, 35, 7, 42),\\
 (28, 63, 35, 7, 42, 14, 49),&
 (35, 7, 42, 14, 49, 21, 56),&
 (42, 14, 49, 21, 56, 28, 63),\\
 (49, 21, 56, 28, 63, 35, 7),&
 (56, 28, 63, 35, 7, 42, 14),&
 (63, 35, 7, 42, 14, 49, 21)
  \end{array}
  \right\}
\]
We thus compute the number of $(\Krewplus)^k$-invariant subwords as
\[
  5\binom{a+1}{1} + 4\binom{a}{1} = 9a+5,
\]
where the first $5$ come from the $5$ factorisations with ascents $i_1 < 36$, while the second $4$ come from the $4$ factorisations with descents $i_1 > 36$.
With $m' = 3$, we obtain $m = 7a+3$, which, by \Cref{eq2:E6.2}, leads to
\[
  \frac{9m+8}{7} = \frac{63a+35}{7} = 9a+5,
\]
as desired.

\subsubsection*{\sc\bf Case $E_8$}
The degrees are $2,8,12,14,18,20,24,30$, and $(m+1)h-2 = 30m+28 = k
\cdot \ell$. We hence have 
\begin{multline}
\mCatplus(E_8;q)=\frac
{[30m+28]_q\, [30m+22]_q\, [30m+18]_q\, [30m+16]_q} 
{[30]_q\, [24]_q\, [20]_q\, [18]_q}\\
\times
\frac 
{[30m+12]_q\, [30m+10]_q\, [30m+6]_q\, [30m]_q} 
{[14]_q\, [12]_q\, [8]_q\, [2]_q} .\label{eq2:E8}
\end{multline}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq2:E8.1}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\mCatplus(E_8),
&\text{if }\ell \in \{1,2\},\\
\label{eq2:E8.2}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\tfrac {(15m+14)(15m+8)(5m+2)m}{64},
&\text{if }m\equiv0\text{ (mod $2$)},\ \ell=4,\\
\label{eq2:E8.3}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\tfrac{(15m+14)(5m+2)}{16},
&\text{if }m\equiv2\text{ (mod $4$)},\ \ell=8,\\
\label{eq2:E8.4}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=0,
&\text{otherwise.}
\end{align}
\reseteqn

We may assume that $\ell$ is even.
In this case $\ell/2$ divides~$n=8$, implying $\ell \in \{2,4,8,16\}$.

The case $\ell = 2$ is computed using all ascent-descent configurations of the $37{,}968{,}750$ factorisations of~$c$.

The case $\ell = 4$ is given by the parameters
\begin{align*}
  y = 2,\quad m' = 0,\quad \ifix =  57.
\end{align*}
There are $6{,}750$ factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}.
Their ascent-descent sequences imply that the number of $(\Krewplus)^k$-invariant subwords as
\[
  627\binom{a+3}{4} + 3784\binom{a+2}{4} + 2251\binom{a+1}{4} + 88\binom{a}{4} = \frac{1}{4}(15a + 7)(15a + 4)(5a + 1)a.
\]
With $m' = 0$, we obtain $m = 2a$, which, by \Cref{eq2:E8.2}, leads to
\begin{align*}
  \frac{(15m+14)(15m+8)(5m+2)m}{64}  &= \frac {(30a+14)(30a+8)(10a+2)2a}{64} \\
                                     &= \frac {(15a+7)(15a+4)(5a+1)a}{4},
\end{align*}
as desired.

The case $\ell = 8$ is given by the parameters
\begin{align*}
  y = 2,\quad m' = 2,\quad \ifix =  89.
\end{align*}
There are $150$ factorizations of the bipartite Coxeter element that satisfy the properties in \Cref{prop:invariantsittingdetails}.
Their ascent-descent sequences imply that the number of $(\Krewplus)^k$-invariant subwords as
\[
  33\binom{a+2}{4} + 109\binom{a+1}{4} + 8\binom{a}{4} = (15a + 11)(5a + 3).
\]
With $m' = 2$, we obtain $m = 4a+2$, which, by \Cref{eq2:E8.3}, leads to
\begin{align*}
  \frac{(15m+14)(5m+2)}{16}  &= \frac {(60a+44)(20a+12)}{16} \\
                             &= (15a+11)(5a+3),
\end{align*}
as desired.

\subsection{Cyclic sieving for $\Krewplustilde$: the exceptional groups}
\label{sec:siev1}

The computations for $\Krewplustilde$ are completely analogous to the computations for $\Krewplus$ in those types for which $\psi \equiv \one$ holds.
We therefore do not repeat the computations here but only present the confirmed cardinalities.
The only case for which we indeed need a new construction is the case of $E_6$ with a $5$-fold symmetry.
This case is discussed in detail below.

\subsubsection*{\sc\bf Case $H_3$}
The degrees are $2,6,10$, and $mh-2 = 10m-2 = k\cdot\ell$.
We hence have
\begin{align}
\mCatplus(H_3;q)=\frac 
{[10m+8]_q\, [10m+4]_q\, [10m]_q} 
{[10]_q\, [6]_q\, [2]_q} .\label{eq:H3}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:H3.1}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=\mCatplus(H_3),
&\text{if } \ell\in\{1,2\},\\
\label{eq:H3.2}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=\tfrac {5m+2}3,
&\text{if } m\equiv 2\text{ (mod $3$)},\ \ell\in\{3,6\},\\
\label{eq:H3.3}
\lim_{q\to\zeta}\mCatplus(H_3;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\subsubsection*{\sc\bf Case $H_4$}
The degrees are $2,12,20,30$, and $mh-2 = 30m-2 = k\cdot\ell$.
We hence have
\begin{align}
\mCatplus(H_4;q)=\frac 
{[30m+28]_q\, [30m+18]_q\, [30m+10]_q\, [30m]_q} 
{[30]_q\, [20]_q\, [12]_q\, [2]_q} .\label{eq:H4}
\end{align}
Let $\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:H4.1}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=\mCatplus(H_4),
&\text{if }\ell\in\{1,2\},\\
\label{eq:H4.2}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=\tfrac {(5m+3)(3m+1)}4,
&\text{if }m\equiv1\text{ mod $2$},\ \ell = 4,\\
\label{eq:H4.3}
\lim_{q\to\zeta}\mCatplus(H_4;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\subsubsection*{\sc\bf Case $F_4$}
The degrees are $2,6,8,12$, and $mh-2 = 12m-2 = k\cdot\ell$.
We hence have
\begin{align}
\mCatplus(F_4;q)=\frac 
{[12m+10]_q\, [12m+6]_q\, [12m+4]_q\, [12m]_q} 
{[12]_q\, [8]_q\, [6]_q\, [2]_q} .\label{eq:F4}
\end{align}
Let~$\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:F4.1}
\lim_{q\to\zeta}\mCatplus(F_4;q)&=\mCatplus(F_4),
&\text{if }\ell\in\{1,2\},\\
\label{eq:F4.2}
\lim_{q\to\zeta}\mCatplus(F_4;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\subsubsection*{\sc\bf Case $E_6$}
The degrees are $2,5,6,8,9,12$, and $mh-2 = 12m-2 = k\cdot\ell$.
We hence have
\begin{align}
\mCatplus(E_6;q)=\frac 
{[12m+10]_q\, [12m+7]_q\, [12m+6]_q\, [12m+4]_q\, [12m+3]_q\, [12m]_q} 
{[12]_q\, [9]_q\, [8]_q\, [6]_q\, [5]_q\, [2]_q} .\label{eq:E6}
\end{align}
Let~$\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:E6.1}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=\mCatplus(E_6),
&\text{if }\ell=1,\\
\label{eq:E6.2}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=\tfrac {(6m+5)(2m+1)(3m+1)m}2,
&\text{if }\ell=2,\\
\label{eq:E6.3}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=\tfrac {12m+3}5,
&\text{if }m\equiv1\text{ (mod $5$)},\ \ell=5,\\
\label{eq:E6.4}
\lim_{q\to\zeta}\mCatplus(E_6;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\subsubsection*{\sc\bf Case $E_7$}
The degrees are $2,6,8,10,12,14,18$, and $mh-2 = 18m-2 = k\cdot\ell$.
We hence have
\begin{multline}
\mCatplus(E_7;q)=\frac 
{[18m+16]_q\, [18m+12]_q\, [18m+10]_q} 
{[18]_q\, [14]_q\, [12]_q}\\
\times
\frac 
{[18m+8]_q\, [18m+6]_q\, [18m+4]_q\, [18m]_q} 
{[10]_q\, [8]_q\, [6]_q\, [2]_q} .\label{eq:E7}
\end{multline}
Let~$\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:E7.1}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\mCatplus(E_7),
&\text{if }\ell\in\{1,2\},\\
\label{eq:E7.2}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\tfrac{(9m+5)(3m+1)}8,
&\text{if }m\equiv1\text{ (mod $2$)},\ \ell=4,\\
\label{eq:E7.3}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\tfrac {9m+3}4,
&\text{if }m\equiv1\text{ (mod $4$)},\ \ell=8,\\
\label{eq:E7.4}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\tfrac {9m+4}5,
&\text{if }m\equiv4\text{ (mod $5$)},\ \ell\in\{5,10\},\\
\label{eq:E7.5}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=\tfrac {9m+6}7,
&\text{if }m\equiv4\text{ (mod $7$)},\ \ell\in\{7,14\},\\
\label{eq:E7.6}
\lim_{q\to\zeta}\mCatplus(E_7;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\subsubsection*{\sc\bf Case $E_8$}
The degrees are $2,8,12,14,18,20,24,30$, and $mh-2 = 30m-2 = k\cdot\ell$.
We hence have
\begin{multline}
\mCatplus(E_8;q)=\frac 
{[30m+28]_q\, [30m+22]_q\, [30m+18]_q\, [30m+16]_q} 
{[30]_q\, [24]_q\, [20]_q\, [18]_q}\\
\times
\frac 
{[30m+12]_q\, [30m+10]_q\, [30m+6]_q\, [30m]_q} 
{[14]_q\, [12]_q\, [8]_q\, [2]_q} . \label{eq:E8}
\end{multline}
Let~$\zeta$ be a primitive $\ell$\th\ root of unity.
The following cases on the right-hand side occur:
\alphaeqn
\begin{align} 
\label{eq:E8.1}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\mCatplus(E_8),
&\text{if }\ell\in\{1,2\},\\
\label{eq:E8.2}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\tfrac {(15m+11)(5m+3)(3m+1)(5m+1)}{64},
&\text{if } m\equiv3\text{ (mod $2$)},\ \ell=4,\\
\label{eq:E8.3}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\tfrac{(15m+11)(5m+1)}{16},
&\text{if }m\equiv3\text{ (mod $4$)},\ \ell=8,\\
\label{eq:E8.4}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=\tfrac {15m+6}7,
&\text{if }m\equiv1\text{ (mod $7$)},\ \ell\in\{7,14\},\\
\label{eq:E8.5}
\lim_{q\to\zeta}\mCatplus(E_8;q)&=1,
&\text{otherwise.}
\end{align}
\reseteqn

\section{Panyushev's analoga on positive nonnesting partitions}
\label{sec:Panyushev}

In this section, we revisit two conjectures of Panyushev~\cite{Pan2008} on antichains in root posets associated to crystallographic Coxeter systems, and discuss possible connections to the positive Kreweras map. We start with recalling some further background. Let $\Delta \subseteq \Phi^+$ be the \high{simple} and \high{positive roots} for the Coxeter systems $(W,\reflS)$. For crystallographic Coxeter systems, define the \defn{root poset} $\rootPoset(W)$ to be the partial order on $\Phi^+$ induced by the cover relations given by $\alpha \prec \beta$ if $\beta - \alpha \in \Delta$. The \defn{nonnesting partitions} for~$W$ are defined by the set of all antichains in $\rootPoset$,
$$ \NN := \big\{ \antichain \subseteq \rootPoset : \antichain \text{ antichain} \big\}.$$
As for non-crossing partitions, there is also a positive variant of nonnesting partitions given by
$$ \NNplus := \big\{ \antichain \subseteq \rootPoset\setminus \Delta : \antichain \text{ antichain} \big\}.$$
These (positive) nonnesting partitions turn out to share many enumerative properties with non-crossing partitions, see~\cite{Arm2006} for a detailed treatment thereof. In particular, we have that
$$ \big| \NN \big| = \big| \NC \big|, \quad \big| \NNplus \big| = \big| \NCplus \big|.$$
Obviously, these equalities call for bijections between both sets, and multiple bijections were constructed in various types~\cite{CDDSY2007,FG2009,RS2010}, the only uniformly described bijection was given in~\cite{AST2010}. But we are still lacking a proper understanding of the connection between both worlds.

One particular avenue to follow --~as explained by Armstrong, Thomas, and the second author~\cite{AST2010}~-- are two maps on $\NC$ and on $\NN$ given by the Kreweras map $\Krew : \NC \longrightarrow \NC$ and the \defn{Panyushev map} $\Pan : \NN \longrightarrow \NN$ which is given by mapping an antichain $\antichain$ to the collection of minimal elements in the root poset among all elements that are not below $\antichain$. In~\cite{AST2010}, a uniformly described bijection between non-crossing and nonnesting partitions is constructed sending the Kreweras map to the Panyushev map. This bijection is uniquely determined (but actually strongly overdetermined) by certain inductive properties on parabolic subgroups. The proof of its existence is given by explicitly constructing it in the various types. Remark that this bijection does not send positive non-crossing partitions to positive nonnesting partitions. Moreover, as mentioned above for the Kreweras map, the Panyushev map does not stabilize positive nonnesting partitions either. But again, there is a similarly defined map $\Panplus : \NNplus \longrightarrow \NNplus$ given by the same rule as for $\Pan$ but considered on the root poset $\rootPoset$ with the simple roots $\Delta$ removed.

Before raising two questions on the connections between positive non-crossing and positive nonnesting partitions, we copy two conjectures on the maps $\Pan$ and $\Panplus$ by Panyushev~\cite[Conjectures~2.1 and~2.3]{Pan2008}, the first of which was proven in~\cite{AST2010} by constructing an equivariant bijection between $\NN$ and $\NC$ as briefly mentioned above.

%Christian: \sc
\begin{theorem}[\sc Armstrong, Stump, Thomas, conjectured by Panyushev]
  Let~$W$ be a finite, crystallographic Coxeter group with Coxeter number~$h$, and let $\Pan$ be the Panyushev map on $\NN$. The order of $\Pan$ is given by
  $$
    \order(\Pan) =
    \begin{cases}
      (m+1)h, & \text{if } \psi \nequiv \one, \\
      (m+1)h/2, & \text{if } \psi \equiv \one.
    \end{cases}
  $$
\end{theorem}
\begin{conjecture}[Panyushev]
  Under the same assumptions as in the previous theorem with $\Panplus$ being the positive Panyushev map on $\NNplus$, the order of $\Panplus$ is given by
  $$
    \order(\Panplus) =
    \begin{cases}
      (m+1)h - 2, & \text{if } \psi|_{\reflR \setminus \reflS} \nequiv \one, \\
      (m+1)h/2 - 1, & \text{if } \psi_{\reflR \setminus \reflS} \equiv \one.
    \end{cases}
  $$  
\end{conjecture}
% \begin{theorem}[Armstrong, Stump, Thomas, conjectured by Panyushev]
%   Let~$W$ be a finite, crystallographic Coxeter group with Coxeter number~$h$, and let $\Pan$ be the Panyushev map on the corresponding root poset $\rootPoset$. Then
%   \begin{enumerate}[(i)]
%     \item $\Pan^{h}$ acts on $\NN$ as the involution induced by $-\wo$. In particular,\\[-10pt]
%     \item $\Pan^{2h}$ acts as the identity on $\NN$.
%   \end{enumerate}
% \end{theorem}
% \begin{conjecture}[Panyushev]
%   Under the same assumptions as in the previous theorem with $\Panplus$ being the positive Panyushev map on the positive root poset $\rootPoset \setminus \Delta$,
%   \begin{enumerate}[(i)]
%     \item $(\Panplus)^{h-1}$ acts on $\NNplus$ as the involution induced by $-\wo$. In particular,\\[-10pt]
%     \item $(\Panplus)^{2h-2}$ acts as the identity on $\NNplus$.
%   \end{enumerate}
% \end{conjecture}

Observe here that in the original conjecture, the restriction of $\psi$ to non-simple reflections was actually missing. The only difference occurs in type $A_2$, where $\psi$ interchanges the two simple reflections $s_1,s_2$ while fixing the third reflection $s_1s_2s_1$. The combinatorial models in types $A$ and $B$ for the bijection in~\cite{AST2010} immediately give that, in these types, the triple
$$\Big( \NNplus, \Panplus, \Cat_+(W;q) \Big)$$
exhibits the cyclic sieving phenomenon. 
This holds as well in the exceptional types,
which was checked using a computer,
thus leaving only type $D$ open. 
Together with \Cref{thm:order} that now raises 
the following two questions.

\begin{question}
  Is there an equivariant bijection between $\NNplus$ and $\NCplus$ similar to the bijection between $\NN$ and $\NC$?
\end{question}

\begin{question}
  What is the connection between the shift functor on the orbit category corresponding to the word $\Q$ and the Panyushev map on positive nonnesting partitions?
\end{question}

We close with the remark that there are as well $m$-extensions $\mNN$ and $\mNNplus$ of (positive) nonnesting partitions. Unfortunately, no extension of the Panyushev map in that direction is known so far. In particular, we still lack an understanding of uniform connections between $m$-divisible non-crossing 
partitions and the $m$-extension of nonnesting partitions.


\begin{comment}

\section{Proof of \Cref{thm:positivedescription}}
\label{sec:positivedescription}

\cs{provide more background, give examples, see if we can change the roles of $w_0$ and $w_m$, make things more explicit}

In this section, we prove \Cref{thm:positivedescription}
and \Cref{cor:unique}. The theorem provides a new description of positive $m$-divisible non-crossing partitions in terms of the $\reflR$-word defined in the introduction.
To this end, we need to recall some background on finite root systems and the connection between non-crossing partitions and facets of the cluster complex as given in~\cite[Section~6.3.3]{PS2011} in the context of subword complexes. We refer to~\cite{PS2011} and the references therein for further background.
As before, let $\wwo(\c)$ be the $\c$-sorting word for the longest element $w_\circ \in W$. Set $\Q = \q_1 \q_2 \cdots \q_{N+n} = \c\wwo(\c)$ be the concatenation of the words $\c$ and $\wwo(\c)$. Moreover, associate to a subword $I = \q_{i_1} \cdots \q_{i_n} \subseteq Q$ and a position $1 \leq k \leq N+n$ of a letter in $\c\wo(\c)$ the root $\Root(I,k)$ as
$$\Root(I,k) := \q_1 \cdots \widehat\q_{i_1} \cdots \widehat\q_{i_2} \cdots \widehat\q_{i_\ell} \cdots \q_{k-1}(\alpha_{\q_k}) \in \Phi,$$
where the letters $\q_{i_1},\ldots,\q_{i_\ell} \in I$ with $i_{\ell+1} \geq k$ are omitted within the product $\q_1 \cdots \q_{k-1}$.
Then there is a bijective correspondence between non-crossing partitions $w \in \NC$ and subwords $I = \q_{i_1} \cdots \q_{i_n} \subseteq \Q$ of length $n$ such that the product of all reflection $s_{\Root(I,i_1)}, \ldots, s_{\Root(I,i_n)}$ is the Coxeter element~$c$,
$$s_{\Root(I,i_1)} \cdots s_{\Root(I,i_n)} = c.$$
Denote by $\subwordComplex$ the collection of all subwords $I$ of length $n$ with this property.

\begin{remark}
  It was shown in~\cite{CLS2011} that the simplicial complex defined by $\subwordComplex$ is isomorphic to the cluster complex of type~$W$ defined by Fomin and Zelevinsky in~\cite{FZ2002}. The \defn{root configuration} $\{ \Root(I,k) : \q_k \in I \}$ of a facet $I \in \subwordComplex$ carries the information of the exchange graph of the cluster complex. It was moreover shown in~\cite{PS2011} that the root configuration provides as well the geometry of the generalized~$c$-associahedron as defined in~\cite{HLT2008}.
\end{remark}

It was shown in the proof of~\cite[Thm.~6.24]{PS2011} \cs{referenz in der neuen version} that, up to transpositions of consecutive commuting reflections in $s_{\Root(I,i_1)}, \ldots, s_{\Root(I,i_n)}$, there is an index $\ell$ for which
$$
  \Root(I,i_k) \in
  \begin{cases}
    \Phi^+ &\text{if}\ k < \ell, \\
    \Phi^- &\text{if}\ k \geq \ell.
  \end{cases}
$$
Moreover, the bijection $\facetNcp : \subwordComplex \tilde\longrightarrow \NC$ is defined by sending $I \in \subwordComplex$ to $(w_0,w_1) \in \NC$ given by
\begin{align*}
  w_0 &= s_{\Root(I,i_\ell)}\cdots s_{\Root(I,i_n)}, \\
  w_1 &= (c^{-1} s_{\Root(I,1)} c) \cdots (c^{-1} s_{\Root(I,i_{\ell-1})} c).
\end{align*}
Moreover, these products are exactly the canonical reduced $\reflR$-words $\Rword(w_0)$ and $\Rword(w_1)$.
Finally, it is well known and simple to check that $(w_0,w_1) \in \NCplus$ if and only if $\facetNcp^{-1}(\in \subwordComplex$ does not contain any letter within the initial $\c$.
We are now in the position to prove the alternative description of positive $m$-divisible non-crossing partitions.
\begin{proof}[Proof of \Cref{thm:positivedescription}]
  Let $(w_0,w_1,\ldots,w_m) \in \mNC$ and let $w = w_0\cdots w_{m-1} \in \NC$. Then $(w_0,w_1,\ldots,w_m) \in \mNCplus$ if and only if $(w,w_m) \in \NCplus$ if and only if $\facetNcp^{-1}(w,w_{m}) \cap \{1,\ldots,n\} = \emptyset$ if and only if $w_m = w_m^L$, as desired.
\end{proof}

\section{The Kreweras map on positive m-divisible non-crossing partitions}
\label{sec:representation theory}

In this section, we provide a combinatorial model for the positive Kreweras map coming from quiver representation theory. We will then use this model to uniformly compute its order.

It is shown in~\cite[Remark~7.6]{CLS2011} that, up to commutations,
\begin{align}
  \c^h = \wwo(\c)\ \psi(\wwo(\c)), \label{eq:ctotheh}
\end{align}
and, in particular, that the bi-infinite words
$$ \widetilde \Q := \dots \ \c \ \c \ \c \ \dots = \dots \ \wwo(\c)\ \psi(\wwo(\c)) \ \wwo(\c)\ \psi(\wwo(\c)) \ \dots$$
coincide up to commutations, and do, again up to commutations, {\bf not} depend on the particular choice of the Coxeter element~$c$. This induced two operations on letters in~$\widetilde \Q$. The first is the \defn{Auslander--Reiten translate} $\ARtranslate$ sending a letter $c_i$ in the $j$\th\ copy of $\c$ to the same letter in the $(j-1)$\st\ copy. The second is the \defn{shift} $\shift$ sending a letter $s_{k_i}$ in a copy of $\wwo(\c)$ to the corresponding letter $\psi(s_{k_i})$ in the next copy of $\psi(\wwo(\c))$ to the right.

\begin{remark}
  This bi-infinite word appeared in various places before. In quiver representation theory it corresponds to the \high{repetition quiver} playing for example an important role in the theory of \high{(higher) cluster categories}, see e.g.~\cite{BMRRT2006,Tho2007,BRT2012}. It was also combinatorially studied in~\cite{CLS2011,IS2010}.
\end{remark}

To see positive $m$-divisible non-crossing partitions using this setting, we next identify letters within $\widetilde \Q$ as $\Q := \widetilde \Q \big/ \big( s \sim \shift^{m+1}\ARtranslate(s) \big)$. Choosing a particular Coxeter element corresponds to choosing a fundamental domain in $\widetilde \Q$ for $\Q$ which looks like $ \Q = \Q_1 \dots \Q_m \Q_{m+1}$, where
\begin{align}
\Q_i =
  \begin{cases}
     \wwo(\c), & \text{if } i\leq m \text{ odd}, \\
     \psi(\wwo(\c)), & \text{if } i\leq m \text{ even}, \\
     \wwo(\c)\setminus\psi(\c), & \text{if } i = m+1 \text{ odd}, \\
     \psi(\wwo(\c))\setminus\c, & \text{if } i = m+1 \text{ even},
  \end{cases} \label{eq:Qi}
\end{align}
where we removed in the third (and fourth, respectively) case, the suffix $\psi(\c)$ or $\c$ from $\wwo(\c)$, or from $\psi(\wwo(\c))$, respectively.

We can now map $(w_0,w_1,\dots,w_m) \in \mNCplus$ to a collection of letters in $\Q$ by mapping $w_i$ to the letters corresponding to the reflections in its associated reduced $\reflR$-word within $\Q_i$. Here, \Cref{thm:positivedescription} ensures that $w_m$ does indeed not use letters within the final $\psi(\c)$ or $\c$ respectively. There is now an explicit representation theoretic description of the image of $\mNCplus$ among all subwords in $\Q$. Since we do not need this description here, we refer to~\cite{BRT2012} for further details.

\begin{proposition}
  The positive Kreweras map on $\mNCplus$ corresponds to the shift $\shift$ on $\Q$ induced by the shift on $\widetilde \Q$.
\end{proposition}
\begin{proof}
  \cs{tba}
\end{proof}

We are now in the position to compute the order of $\Krewplus$ on $\mNCplus$.
\begin{proof}[Proof of \Cref{thm:order}]
  By the definition of $\Q$ and \eqref{eq:ctotheh}, we have that
  $$ \ARtranslate^{-1} = \shift^{m+1}, \quad \ARtranslate^h = \shift^{-2}.$$
  Thus, we immediately obtain that $\shift^{h(m+1)-2}$ acts trivially on letters in $\Q$. In the case of $\psi \equiv \one$, we moreover have that $\ARtranslate^{h/2} = \shift^{-1}$, which gives that in this case, even $\shift^{h(m+1)/2-1}$ acts trivially on letters in $\Q$.

  It remains to show that for any two letters in $\Q$, there is a positive $m$-divisible non-crossing partition whose representation on letters in $Q$ contains exactly one of the two. It is simple to see that this is the case if we are in rank bigger than $2$. Thus, the only special cases are the dihedral groups for which $\psi \equiv \one$. For those, the situation is fine if $m>1$, otherwise, we have a symmetry of order $2$. This completes the proof.
\end{proof}

\appendix

\section*{Appendix: Polynomiallity of the positive Fu\ss-Catalan numbers}

\setcounter{subsection}{0}%
\global\def\thesubsection{\mbox{A.\arabic{subsection}}}
\setcounter{theorem}{0}%
\global\def\thetheorem{\mbox{A.\arabic{theorem}}}
\setcounter{equation}{0}%
\global\def\theequation{\mbox{A.\arabic{equation}}}

\subsection{Type~A}
\subsection{Type~B}
\subsection{Type~D}
\subsection{Exceptional types}

The following auxiliary results, providing special expressions
which turn out to be polynomials with non-negative coefficients,
are taken from \cite[Cor.~1 and Lemma~2]{KrMuAD}.
They are reproduced here for the convenience of the reader.

\begin{lemma} \label{cor:A}
Let $a$ and $b$ be coprime positive integers, and let $\ga$ be an integer
with $\ga\ge (a-1)(b-1)$. Then the expression
$$
\frac {[\ga]_q\,[ab]_q} {[a]_q\,[b]_q}
$$
is a polynomial in $q$ with non-negative
integer coefficients.
\end{lemma}

\begin{lemma} \label{lem:B}
Let $\al$ and $\be$ be positive integers.
The following expressions are polynomials in $q$ with non-negative
integer coefficients:

\begin{enumerate}
\item[(a)]%!!
$
\left[\al\right]_{q^3}\left[\be\right]_{q^4}
\frac {\left[72\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[8\right]_q\left[9\right]_q\left[12\right]_q}
$\quad 
for $\al\ge 6$ and $\be\ge 8$;
\item[(b)]$ %lem:G 
\left[\al\right]_{q}\left[\be\right]_{q^4}
\frac {\left[15\right]_q}
{\left[3\right]_q\left[5\right]_q}
\frac {\left[72\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[8\right]_q\left[9\right]_q\left[12\right]_q}
$\quad 
for $\al\ge 26$ and $\be\ge8$;
\item[(c)]$ %lem:C
\left[\al\right]_{q^3}\left[\be\right]_{q^4}
\frac {\left[90\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[5\right]_q\left[6\right]_q\left[9\right]_q}
$\quad 
for $\al\ge 18$ and $\be\ge3$;
\item[(d)]$ %lem:F !!
\left[\al\right]_{q}\left[\be\right]_{q^3}
\frac {\left[90\right]_q\left[3\right]_q}
{\left[5\right]_q\left[6\right]_q\left[9\right]_q}
$\quad 
for $\al\ge 20$ and $\be\ge18$;
\item[(e)]$ %lem:D
\left[\al\right]_{q}
\frac {\left[15\right]_q}
{\left[3\right]_q\left[5\right]_q}
\frac {\left[12\right]_{q^3}}
{\left[3\right]_{q^3}\left[4\right]_{q^3}}
$\quad 
for $\al\ge 26$;
\item[(f)]$ %lem:E!!
\left[\al\right]_{q}
\frac {\left[15\right]_q}
{\left[3\right]_q\left[5\right]_q}
\frac {\left[6\right]_{q^3}}
{\left[2\right]_{q^3}\left[3\right]_{q^3}}
$\quad 
for $\al\ge 14$;
\item[(g)]$ %lem:H
\left[\al\right]_{q}\left[\be\right]_{q^2}
\frac {\left[84\right]_q\left[2\right]_q}
{\left[4\right]_q\left[6\right]_q\left[7\right]_q}
$\quad 
for $\al\ge 30$ and $\be\ge20$;
\item[(h)]$ %lem:I
\left[\al\right]_{q}\left[\be\right]_{q}
\frac {\left[105\right]_q}
{\left[3\right]_q\left[5\right]_q\left[7\right]_q}
$\quad 
for $\al\ge 24$ and $\be\ge68$;
\item[(i)]$ %lem:J
\left[\al\right]_{q}\left[\be\right]_{q}
\frac {\left[70\right]_q}
{\left[2\right]_q\left[5\right]_q\left[7\right]_q}
$\quad 
for $\al\ge 24$ and $\be\ge34$;
\item[(j)]$ %lem:61015 !!
\left[\al\right]_{q^2}\left[\be\right]_{q^5}
\frac {\left[30\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[6\right]_q\left[10\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 4$ and $\be\ge2$;
\item[(k)]$ %lem:61015B
\left[\al\right]_{q}\left[\be\right]_{q^5}
\frac {\left[14\right]_q}
{\left[2\right]_q\left[7\right]_q}
\frac {\left[30\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[6\right]_q\left[10\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 14$ and $\be\ge2$;
\item[(l)]$ %lem:61015C
\left[\al\right]_{q}\left[\be\right]_{q^2}
\frac {\left[35\right]_q}
{\left[5\right]_q\left[7\right]_q}
\frac {\left[30\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[6\right]_q\left[10\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 32$ and $\be\ge12$;
\item[(m)]$ %lem:101215
\left[\al\right]_{q^2}\left[\be\right]_{q^5}
\frac {\left[60\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[10\right]_q\left[12\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 16$ and $\be\ge2$;
\item[(n)] %lem:101215B
$\left[\al\right]_{q}\left[\be\right]_{q^2}
\frac {\left[35\right]_q}
{\left[5\right]_q\left[7\right]_q}
\frac {\left[60\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[10\right]_q\left[12\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 56$ and $\be\ge4$;
\item[(o)]$ %lem:101215C
\left[\al\right]_{q}\left[\be\right]_{q^5}
\frac {\left[14\right]_q}
{\left[2\right]_q\left[7\right]_q}
\frac {\left[60\right]_q\left[2\right]_q\left[3\right]_q\left[5\right]_q}
{\left[10\right]_q\left[12\right]_q\left[15\right]_q}
$\quad 
for $\al\ge 38$ and $\be\ge2$;
\item[(p)]$ %lem:679
\left[\al\right]_{q}\left[\be\right]_{q^3}
\frac {\left[126\right]_q\left[3\right]_q}
{\left[6\right]_q\left[7\right]_q\left[9\right]_q}
$\quad 
for $\al\ge 30$ and $\be\ge26$;
\item[(q)]$ %lem:7912
\left[\al\right]_{q}\left[\be\right]_{q^3}
\frac {\left[252\right]_q\left[3\right]_q}
{\left[7\right]_q\left[9\right]_q\left[12\right]_q}
$\quad 
for $\al\ge 66$ and $\be\ge54$;
\item[(r)]$ %lem:4710
\left[\al\right]_{q}\left[\be\right]_{q^2}
\frac {\left[140\right]_q\left[2\right]_q}
{\left[4\right]_q\left[7\right]_q\left[10\right]_q}
$\quad 
for $\al\ge 54$ and $\be\ge34$.
\end{enumerate}
\end{lemma}

For $W=I_2(a)$, the degrees are $2,a$, and hence
\begin{align*}
\Cat^m(I_2(a);q)&=\frac {[am]_q\,[am+a-2]_q} 
{[2]_q\,[2]_q\,[a]_q}\\
&=\begin{cases}
\left[m\right]_{q^{a}}\,
\left[\tfrac {am+a-2} {2}\right]_{q^2}\,
,&\text{if $m$ is odd,}\\
\left[\tfrac {m} {2}\right]_{q^{2a}}\,
\left[am+a-2\right]_{q}\,
,&\text{if $m$ is even,}
\end{cases}
\end{align*}
which are polynomials in $q$ with 
non-negative integer coefficients in both cases.

For $W=H_3$, the degrees are $2,6,10$, and hence
\begin{align*}
\Cat^m(H_3;q)&=\frac {[10m]_q\,[10m+4]_q\,[10m+8]_q} 
{[2]_q\,[6]_q\,[10]_q}\\
&=\begin{cases} 
\left[\tfrac {m} {3}\right]_{q^{30}}\,
\left[5m+2\right]_{q^2}\,
\left[5m+4\right]_{q^2}\,
\frac {[15]_{q^2}} {[3]_{q^2}\, [5]_{q^2}},&\text{if $m\equiv0$ (mod 3),}\\
\left[m\right]_{q^{10}}\,
\left[5m+2\right]_{q^2}\,
\left[\tfrac {5m+4} {3}\right]_{q^6}
,&\text{if $m\equiv1$ (mod 3),}\\
\left[m\right]_{q^{10}}\,
\left[\tfrac {5m+2} {3}\right]_{q^6}\,
\left[5m+3 \right]_{q^2}\,
,&\text{if $m\equiv2$ (mod 3),}
\end{cases}
\end{align*}
which, by \Cref{cor:A}, are polynomials in $q$ with 
non-negative integer coefficients in all cases.


For $W=H_4$, the degrees are $2,12,20,30$, and hence
$$
\Cat^m(H_4;q)=\frac {[30m]_q\,[30m+10]_q\,[30m+18]_q\,[30m+28]_q}
{[2]_q\,[12]_q\,[20]_q\,[30]_q}.
$$
If $m$ is even, then we have
$$
\Cat^m(H_4;q)=
\left[\frac {m} {2}\right]_{q^{60}}\,
\left[3m+1\right]_{q^{10}}\,
\left[5m+3\right]_{q^{6}}\,
\left[\tfrac {15m+14}2\right]_{q^{4}}\,
\frac {[30]_{q^2}\left[2\right]_{q^2}
\left[3\right]_{q^2}\left[5\right]_{q^2}} 
{[6]_{q^6}\left[10\right]_{q^2}\left[15\right]_{q^2}},
$$
which, by \Cref{lem:B}(j), is a polynomial in $q$ with 
non-negative integer coefficients. 
On the other hand,
if $m$ is odd, then we may write
\begin{align*}
\Cat^m(H_4;q)&=
\left[m\right]_{q^{30}}\,
\left[\tfrac{3m+1}2\right]_{q^{20}}\,
\left[\tfrac {5m+3} {2}\right]_{q^{12}}\,
\left[15m+14\right]_{q^{2}},
\end{align*}
which is manifestly a polynomial in $q$ with 
non-negative integer coefficients. 

For $W=F_4$, the degrees are $2,6,8,12$, and hence
\begin{align*}
\Cat^m(F_4;q)&=\frac {[12m]_q\,[12m+4]_q\,[12m+6]_q\,[12m+10]_q}
{[2]_q\,[6]_q\,[8]_q\,[12]_q}\\
&=
\begin{cases} 
\left[\frac {m} {2}\right]_{q^{24}}\,
\left[3m+1\right]_{q^{4}}\,
\left[2m+1\right]_{q^{6}}\,
\left[6m+5\right]_{q^{2}}\,
\dfrac {[6]_{q^4}} {[2]_{q^4}\,[3]_{q^4}}
,&\text{if $m$ is even,}\\
\left[m\right]_{q^{12}}\,
\left[\tfrac {3m+1} {2}\right]_{q^{8}}\,
\left[2m+1\right]_{q^{6}}\,
\left[6m+5\right]_{q^2}\,
,&\text{if $m$ is odd,}\\
\end{cases}
\end{align*}
which in both cases is a polynomial in $q$ with 
non-negative integer coefficients; in the first case this is
due to \Cref{cor:A}.

For $W=E_6$, the degrees are $2,5,6,8,9,12$, and hence
$$
\mCatplus(E_6;q)=\frac {[12m]_q\,[12m+3]_q\,[12m+4]_q\,[12m+6]_q\,
[12m+7]_q\,[12m+10]_q}
{[2]_q\,[5]_q\,[6]_q\,[8]_q\,[9]_q\,[12]_q}.
$$
If $m\equiv
0
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[\tfrac {m} {6}\right]_{q^{72}}\,\left[4m+1\right]_{q^3}\,
\left[3m+1\right]_{q^4}\,
\frac {\left[72\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[8\right]_q\left[9\right]_q\left[12\right]_q}\\
\times
\left[2m+1\right]_{q^6}\,
\left[12m+7\right]_q\,\left[\tfrac {6m+5} {5}\right]_{q^{10}}
\frac {\left[10\right]_q} {\left[2\right]_q\left[5\right]_q}
,
\end{multline*}
which, by \Cref{cor:A} and \Cref{lem:B}(a), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
1
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} 
 \left[ \tfrac{1}{5} (4 m+1)\right]_{q^{ 15}} \left[ 
  \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ \tfrac{1}{3} (2 m+1)
  \right]_{q^{ 18}} 
 \frac {\left[ 15\right]_{q}} 
   {\left[ 3\right]_{q} \left[ 5\right]_{q}}
 \frac {\left[ 6\right]_{q^{ 3}} } 
    { \left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}}\\
\times
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}},
\end{multline*}
which, by \Cref{lem:B}(f), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
2
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 9}} 
 \left[ 3 m+1\right]_{q^{ 4}} \left[ 
   \tfrac{1}{5} (2 m+1)\right]_{q^{ 30}} \frac {\left[ 30 
   \right]_{q} } { \left[ 5\right]_{q} \left[ 6 
   \right]_{q}} \\
\times\left[ 12 m+7\right]_{q} \left[ 
  6 m+5\right]_{q^2}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
3
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{3}\right]_{q^{ 36}} \frac {\left[ 12 
   \right]_{q^{ 3}} } { \left[ 3\right]_{q^{ 3}} \left[ 4 
   \right]_{q^{ 3}}} \left[ 4 m+1\right]_{q^3} \left[ 
   \tfrac{1}{10} (3 m+1)\right]_{q^{ 40}} \frac {\left[ 40 
   \right]_{q} } { \left[ 5\right]_{q} \left[ 8 
   \right]_{q}} \left[ 2 m+1\right]_{q^{ 6}} \\
\times
\left[ 
  12 m+7\right]_{q} \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
4
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ 4 m+1\right]_{q^3} \left[ 
  3 m+1\right]_{q^{ 4}} \left[ \tfrac{1}{3} (2 m+1) 
   \right]_{q^{ 18}} \frac {\left[ 6\right]_{q^{ 3}} } { 
  \left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}} \\
\times\left[ 
  \tfrac{1}{5} (12 m+7)\right]_{q^{ 5}} \left[ 6 m+5 
  \right]_{q^2}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
5
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 
  9}} \left[ \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\
\times\left[ \tfrac{1}{2} (12 m+7)\right]_{q^{ 2}} 
 \left[ \tfrac{2}{5} (6 m+5)\right]_{q^{ 5}}
,\end{multline*}
which is manifestly a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
6
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac {m} {6}\right]_{q^{ 72}} 
\frac {\left[72 \right]_{q} \left[ 3\right]_{q} \left[ 4\right]_{q}} 
    {\left[ 8\right]_{q} \left[ 9\right]_{q} \left[ 12\right]_{q}}
 \left[ \tfrac {1} {5}(4 m+1)\right]_{q^{ 15}} 
\frac {\left[ 15\right]_{q}} 
    {\left[ 3\right]_{q} \left[ 5\right]_{q}}\\ \times
 \left[ 3 m+1\right]_{q^{ 4}} 
 \left[ 2 m+1\right]_{q^{ 6}} 
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5   \right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A} and \Cref{lem:B}(a), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
7
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} 
 \left[ 4 m+1\right]_{q^{ 3}} \left[ \tfrac{1}{2} (3 m+1) 
  \right]_{q^{ 8}} \left[ \tfrac{1}{15} (2 m+1)\right]_{q^{ 90}} 
  \frac {\left[ 90\right]_{q} \left[ 3\right]_{q}} { \left[ 5 
   \right]_{q} \left[ 6\right]_{q} \left[ 9\right]_{q}} \\ \times
 \left[ 12 m+7 
  \right]_{q} \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{lem:B}(d), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
8
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 9}} 
 \left[ \tfrac{4}{5} (3 m+1)\right]_{q^{ 5}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\ \times \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
9
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{3}\right]_{q^{ 36}} \frac {\left[ 12 
   \right]_{q^{ 3}} } { \left[ 3\right]_{q^{ 3}} \left[ 4 
   \right]_{q^{ 3}}} \left[ 4 m+1\right]_{q^3} \left[ 
  \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ 2 m+1\right]_{q^{ 
  6}}\\ \times \left[ \tfrac{1}{5} (12 m+7)\right]_{q^{ 5}} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
10
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ 4 m+1\right]_{q^3} \left[ 
  3 m+1\right]_{q^{ 4}} \left[ \tfrac{1}{3} (2 m+1) 
   \right]_{q^{ 18}} \frac {\left[ 6\right]_{q^{ 3}} } { 
  \left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}} \\ \times \left[ 
  12 m+7\right]_{q} \left[ \tfrac{1}{5} (6 m+5)\right]_{q^{ 
  10}}
\frac {\left[ 10\right]_{q}} 
    {\left[ 2\right]_{q} \left[ 5\right]_{q}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
11
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} \left[ \tfrac{1}{15} (4 m+1) 
   \right]_{q^{ 45}} \frac {\left[ 45\right]_{q} } { 
  \left[ 5\right]_{q} \left[ 9\right]_{q}} \left[ 
  \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ 2 m+1\right]_{q^{ 
  6}} \\ \times \left[ 12 m+7\right]_{q} \left[ 6 m+5 
  \right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
12
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac {m} {6}\right]_{q^{ 72}} 
 \left[ 4 m+1\right]_{q^{ 3}} 
 \left[ 3 m+1\right]_{q^{ 4}} 
\frac {\left[72\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[8\right]_q\left[9\right]_q\left[12\right]_q}
 \left[ \tfrac {1} {5}(2 m+1)\right]_{q^{ 30}} 
\frac {\left[ 30\right]_{q}}
   {\left[ 5\right]_{q} \left[ 6\right]_{q}}\\ \times
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5   \right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A} and \Cref{lem:B}(a), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
13
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} 
 \left[ 4 m+1\right]_{q^{ 3}} \left[ \tfrac{1}{10} (3 m+1)
   \right]_{q^{ 40}} \frac {\left[ 40\right]_{q} } { 
  \left[ 5\right]_{q} \left[ 8\right]_{q}} \left[ 
  \tfrac{1}{3} (2 m+1)\right]_{q^{ 18}}
\frac {\left[ 6\right]_{q^{ 3}}} 
    {\left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}}\\ \times
 \left[ 12 m+7\right]_{q^{ 
  1}} \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
14
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 9}} 
 \left[ 2 (3 m+1)\right]_{q^{ 2}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\ \times \left[ \tfrac{1}{5} (12 m+7)\right]_{q^{ 5 
  }} \left[ 12 m+10\right]_{q}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
15
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac {m} {3}\right]_{q^{ 36}} 
\frac {\left[ 12\right]_{q^{ 3}}}
   {\left[ 3\right]_{q^{ 3}} \left[ 4\right]_{q^{ 3}}}
 \left[ 4 m+1\right]_{q^{ 3}} \left[ \tfrac{1}{2} (3 m+1) 
  \right]_{q^{ 8}} \left[ 2 m+1\right]_{q^{ 6}} \\ \times
 \left[ 12 m+7\right]_{q} \left[ \tfrac{1}{5} (6 m+5) 
  \right]_{q^{ 10}}
\frac {\left[ 10\right]_{q}} 
    {\left[ 2\right]_{q} \left[ 5\right]_{q}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
16
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{5} (4 m+1)\right]_{q^{ 15}} 
\frac {\left[ 15\right]_{q}}
     {\left[ 3\right]_{q} \left[ 5\right]_{q}}
 \left[ 3 m+1\right]_{q^{4}} \left[ 
   \tfrac{1}{3} (2 m+1)\right]_{q^{ 18}} \frac {\left[ 6 
   \right]_{q^{ 3}} } { \left[ 2\right]_{q^{ 3}} \left[ 3 
   \right]_{q^{ 3}}} \\ \times \left[ 12 m+7\right]_{q} \left[ 
  6 m+5\right]_{q^2}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
17
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} 
 \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 9}} \left[ 
  \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ \tfrac{1}{5} (2 m+1)
  \right]_{q^{ 30}}
\frac {\left[ 30\right]_{q}}
    {\left[ 5\right]_{q} \left[ 6\right]_{q}}\\ \times
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
18
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac {m} {6}\right]_{q^{ 72}} 
 \left[ 4 m+1\right]_{q^{ 3}} 
\frac {\left[ 72\right]_{q} \left[ 3\right]_{q} 
        \left[ 4\right]_{q} }
    {\left[ 8\right]_{q} \left[ 9\right]_{q} 
    \left[ 12 \right]_{q} }
 \left[ \tfrac {1} {5}(3 m+1)\right]_{q^{ 20}} 
\frac {\left[ 20\right]_{q} }
    {\left[ 4\right]_{q} \left[ 5\right]_{q} }
 \left[ 2 m+1\right]_{q^{ 6}} \\ \times
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A} and \Cref{lem:B}(a), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
19
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} 
 \left[ 4 m+1\right]_{q^{ 3}} \left[ \tfrac{1}{2} (3 m+1) 
  \right]_{q^{ 8}} \left[ \tfrac{1}{3} (2 m+1)\right]_{q^{ 18 
  }}
\frac {\left[ 6\right]_{q^{ 3}}}
    {\left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}}\\ \times
 \left[ \tfrac{1}{5} (12 m+7)\right]_{q^{ 5}} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
20
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 9}} 
 \left[ 3 m+1\right]_{q^{ 4}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\ \times \left[ 12 m+7\right]_{q} 
 \left[ \tfrac{1}{5} (6 m+5)\right]_{q^{ 10}}
\frac {\left[ 10\right]_{q} }
     {\left[ 2\right]_{q} \left[ 5\right]_{q} }
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
21
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{3}\right]_{q^{ 36}} \frac {\left[ 12 
   \right]_{q^{ 3}} } { \left[ 3\right]_{q^{ 3}} \left[ 4 
   \right]_{q^{ 3}}} \left[ \tfrac{1}{5} (4 m+1)\right]_{q^{ 15}} 
\frac {\left[ 15\right]_{q} }
     {\left[ 3\right]_{q} \left[ 5\right]_{q} }
 \left[ \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\ \times \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
22
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac {m} {2}\right]_{q^{ 24}} 
 \left[ 4 m+1\right]_{q^{ 3}} 
 \left[ 3 m+1\right]_{q^{ 4}} 
\frac { \left[ 6\right]_{q^{ 4}} }
   { \left[ 2\right]_{q^{ 4}}  \left[ 3\right]_{q^{ 4}} }
 \left[ \tfrac {1} {15}(2 m+1)\right]_{q^{ 90}} 
\frac {\left[90\right]_q\left[3\right]_q} 
{\left[5\right]_q\left[6\right]_q\left[9\right]_q}\\ \times
 \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5 \right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A} and \Cref{lem:B}(d), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
23
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} \left[ \tfrac{1}{3} (4 m+1) 
  \right]_{q^{ 9}} \left[ \tfrac{1}{10} (3 m+1)\right]_{q^{ 40}} 
  \frac {\left[ 40\right]_{q} } { \left[ 5 
   \right]_{q} \left[ 8\right]_{q}} \left[ 2 m+1 
  \right]_{q^{ 6}} \\ \times \left[ 12 m+7\right]_{q} 
 \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
24
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \frac {m} {6}\right]_{q^{ 72}} 
 \left[ 4 m+1\right]_{q^{ 3}} 
 \left[ 3 m+1\right]_{q^{ 4}} 
\frac {\left[72\right]_q\left[3\right]_q\left[4\right]_q} 
{\left[8\right]_q\left[9\right]_q\left[12\right]_q}
 \left[ 2 m+1\right]_{q^{ 6}} \\ \times
 \left[ \tfrac {1} {5}(12 m+7)\right]_{q^{ 5}} 
 \left[ 6 m+5  \right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{lem:B}(a), 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
25
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[  m\right]_{q^{ 12}} 
 \left[ 4 m+1\right]_{q^{ 3}} \left[ \tfrac{1}{2} (3 m+1) 
  \right]_{q^{ 8}} \left[ \tfrac{1}{3} (2 m+1)\right]_{q^{ 18 
  }}
\frac {\left[ 6\right]_{q^{ 3}}}
    {\left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}}\\ \times
 \left[ 12 m+7\right]_{q} \left[ 
  \tfrac{1}{5} (6 m+5)\right]_{q^{ 10}}
\frac {\left[ 10\right]_{q}}
    {\left[ 2\right]_{q} \left[ 5\right]_{q}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
26
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ \tfrac{1}{15} (4 m+1)\right]_{q^{ 45}} 
  \frac {\left[ 45\right]_{q} } { \left[ 5 
   \right]_{q} \left[ 9\right]_{q}} \left[ 3 m+1 
  \right]_{q^{ 4}} \left[ 2 m+1\right]_{q^{ 6}} \\ \times
 \left[ 12 m+7\right]_{q} \left[ 6 m+5\right]_{q^{ 
  2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
27
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{3}\right]_{q^{ 36}} \frac {\left[ 12 
   \right]_{q^{ 3}} } { \left[ 3\right]_{q^{ 3}} \left[ 4 
   \right]_{q^{ 3}}} \left[ 4 m+1\right]_{q^3} \left[ 
  \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ \tfrac{1}{5} (2 m+1)
   \right]_{q^{ 30}} \frac {\left[ 30\right]_{q} } { 
  \left[ 5\right]_{q} \left[ 6\right]_{q}} \\ \times \left[ 
  12 m+7\right]_{q} \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
28
~(\text{mod }30),$ then we have 
\begin{multline*}
\mCatplus(E_6;q)=
\left[ \tfrac{m}{2}\right]_{q^{ 24}} \frac {\left[ 6 
   \right]_{q^{ 4}} } { \left[ 2\right]_{q^{ 4}} \left[ 3 
   \right]_{q^{ 4}}} \left[ 12 m+3\right]_{q} \left[ 
  \tfrac{4}{5} (3 m+1)\right]_{q^{ 5}} \left[ \tfrac{1}{3} (2 m+1)
   \right]_{q^{ 18}} \frac {\left[ 6\right]_{q^{ 3}} } { 
  \left[ 2\right]_{q^{ 3}} \left[ 3\right]_{q^{ 3}}} \\ \times \left[ 
  12 m+7\right]_{q} \left[ 6 m+5\right]_{q^{ 2}}
,\end{multline*}
which, by \Cref{cor:A}, 
is a polynomial in $q$ with 
non-negative integer coefficients.
If $m\equiv
29
~(\text{mod }30),$ then we have 
$$
\mCatplus(E_6;q)=
\left[ m\right]_{q^{ 12}} \left[ \tfrac{1}{3} (4 m+1)\right]_{q^{ 
  9}} \left[ \tfrac{1}{2} (3 m+1)\right]_{q^{ 8}} \left[ 2 m+1 
  \right]_{q^{ 6}} \left[ \tfrac{1}{5} (12 m+7)\right]_{q^{ 5}} 
 \left[ 6 m+5\right]_{q^{ 2}}
,$$
which is manifestly a polynomial in $q$ with 
non-negative integer coefficients.

\end{comment}

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