%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2024}
\articlenumber{53}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

% \newtheorem{thm}{Theorem}
% \newtheorem{lem}{Lemma}

\usepackage{lipsum}

%% define your title in the usual way
\title[Whirling and Rowmotion]{Whirling and rowmotion dynamics on the chain of V's poset}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Matthew Plante\thanks{\href{mailto:matthew.plante@uconn.edu}{matthew.plante@uconn.edu}}\addressmark{1} \and Tom Roby\addressmark{1}}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Connecticut\\
}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
%%% ABSTRACT
\abstract{Given a finite poset $P$, we study the \emph{whirling} action on vertex-labelings of $P$
with the elements $\{0,1,2,\dotsc ,k\}$.  When such labelings are (weakly) order-reversing,
we call them \textit{$k$-bounded $P$-partitions}.  
We give a general equivariant bijection between $k$-bounded $P$-partitions and order
ideals of the poset $P\times [k]$ which conveys whirling to the well-studied \textit{rowmotion} operator.  As an
application, we derive periodicity and homomesy results for rowmotion acting on
the \textit{chain of V's} poset $\VV \times [k]$.  We are able to generalize some of these results to
the more complicated dynamics of rowmotion on $\BB_{n}\times [k]$, where $\BB_{n}$ is the
\textbf{claw poset} with $n$ unrelated elements each covering $\widehat{0}$. 
}
%%% END ABSTRACT
%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{posets,
chain of V's,
dynamical algebraic combinatorics,
homomesy,
$P$-partitions,
rowmotion,
whirling.
}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{rowwhirlFPSAC24.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

%%% Our Additions
\usepackage{amsfonts,latexsym,tikz,tikz-cd,hyperref, wrapfig,float,wasysym}

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{question}[thm]{Question}
\newtheorem{notation}[thm]{Notation}
\newtheorem{observation}[thm]{Observation}
\theoremstyle{definition}
\newtheorem{remark}[thm]{Remark}
\newtheorem{definition}[thm]{Definition}
\newtheorem{ex}[thm]{Example}

%%% TO MAKE PROOFS GO AWAY
\usepackage{comment}
\newif\ifshow
%%\showtrue
\showfalse
\ifshow
  \newenvironment{longproof}{\noindent \textbf{Proof:}}{{\hfill$\square$}}
\else
  \excludecomment{longproof}
\fi
%%\newenvironment{longproof}{begin code}{end code}

\newcommand{\uF}{\breve{F}}
\newcommand{\uS}{\breve{S}}
\newcommand{\cU}{{\mathcal U}}
\newcommand{\na}{\nabla}
\newcommand{\rhoh}{\rho_J}
\newcommand{\tauh}{\widehat{\tau}}
\newcommand{\hchi}{\widehat\chi}
\newcommand{\htau}{\tau}
\newcommand{\cTh}{\widehat{\cT}}
\newcommand{\chih}{\widehat{\chi}}
\newcommand{\chib}{\ol{\chi}}
\newcommand{\cOb}{\ol{\cO}}
\newcommand{\bba}{\ol{b}}
\newcommand{\VV}{{\sf V}}
\newcommand{\NN}{\mathbb{N}}

\newcommand{\cc}{ {\sf e} }
\newcommand{\oc}{ {\sf b} }
\newcommand{\BB}{ {\sf C} }
\newcommand{\arho}{\rho_A}
\newcommand{\jrho}{\rho}
\newcommand{\up}{\nabla}
\newcommand{\down}{\Delta}

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  \hspace{0.2cm}
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\newcommand{\tre}{\textcolor{red}}
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\newcommand{\twr}[1]{\todo[size=\small, linecolor=blue, backgroundcolor=orange!20!white,inline,
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

We connect the well-studied operation of \textit{rowmotion} on the order ideals of a finite poset
with the less familiar \textit{whirling} action on $P$-partitions with bounded labels.  One
of our main results is an equivariant bijection that carries one to the other for any finite
poset $P$.  We then leverage this to study the rowmotion action on the ``chain of V's'' poset $\VV_{k}:=\VV \times [k]$
(a 3-element V-shaped poset cross a finite chain, see Figure~\ref{fig:vee}), which has surprisingly good dynamical properties.
We also generalize this to the case where we replace $\VV$ with a \textbf{$n$-claw}, a poset with a single minimal
element covered by exactly $n$ incomparable elements. In both cases we obtain both 
periodicity results and homomesy.  

Let $P$ be a finite poset, and $\mathcal{J}\left(  P\right)$ be the set of
order ideals of $P$. (For basic poset definitions, we refer the reader to 
Stanley~\cite[Ch.~3]{Sta11}.)
\emph{Combinatorial rowmotion} is an invertible map
$\rho:\mathcal{J}(P)\rightarrow\mathcal{J}(P)$ which takes each ideal
$I\in\mathcal{J}(P)$ to the order ideal generated by the minimal elements of
the complement of $I$ in $P$. The periodicity of this map on products of chains was first
studied by Brouwer and Schrijver~\cite{BS74},
and Cameron and Fon-der-Flaass~\cite{CF95}.  Later Striker and
Williams~\cite{SW12} considered it as one element of the ``toggle group'' of a
poset and related it to a kind of ``promotion'' operator on order ideals.  Around the same
time, Armstrong, Stump, and Thomas~\cite{ast} studied rowmotion on
\emph{root posets}, relating it to ``Kreweras complementation'' on noncrossing partitions, and used this to prove a
conjecture of Panyushev about the equality of the average cardinality of antichains for each
rowmotion orbit.  

Propp and Roby~\cite{PR13} noticed that this conjecture was merely one
instance of a much broader phenomenon which they dubbed \emph{homomesy}. Given
a finite set $S$, a \textquotedblleft statistic\textquotedblright%
\ $f:S\rightarrow\mathbb{C}$, and an invertible map $\varphi$ on $S$, we call
$f$ \emph{homomesic} if the average value of $f$ is the same for every
$\varphi$-orbit $\mathcal{R}$, i.e., $\displaystyle\frac{1}{\#\mathcal{R}}%
\sum_{x\in\mathcal{R}}f(x)=c$, where $c$ is a constant not dependent on the
choice of orbit $\mathcal{R}$. The confluence of all this work was the
beginning of \emph{dynamical algebraic combinatorics} as a distinct area
within algebraic combinatorics (with antecedents going back to the
Robinson--Schensted--Knuth correspondence and related operations on Young
tableaux such as promotion, evacuation, and cyclage).  In the past decade, the subfield has
grown in a number of directions, and the study of rowmotion has been of continuing interest.
For more background information, see
the survey articles of Hopkins~\cite{Hopkins-order}, Roby~\cite{robydac}, and Striker~\cite{striker-dac}. 

Cameron and Fon-der-Flaass~\cite{CF95} were the first to describe rowmotion as a product of
involutions called \textit{toggles}, as detailed in Section~\ref{ss:toggles}.  A natural
generalization of toggling at a poset element $x$ is ``\textit{whirling} at $x$,'' which cycles
the label at $x$ among $j$ possible values.  (Toggles are the case when $j=2$.)  Joseph,
Propp, and Roby defined these and the operation of \textit{whirling} on sets of functions
between finite sets, obtaining various homomesy results for various classes of functions
(injective, surjective, etc.)~\cite{JPR18}. This is described in
Section~\ref{sec:whirling}.  

A bijective function $f:P \to [p]$ (with $\# P = p$) such that $f(x)<f(y)$ 
whenever %%%if and only if (WRONG)
$x<_P y$ is called a \emph{linear extension}.  We denote by $\mathcal{L}(P)$ the set of
\emph{all linear extensions}
of $P$; its cardinality, $e(P)$, is an important numerical invariant of a poset. 
Its refinement, the \emph{order polynomial} $\Omega_{P}(k)$, counts the number of $k$-bounded
$P$-partitions.  For some special posets $P$, mainly ones connected with Lie theory (root
and minuscule posets) and those of partition or shifted shapes, product formulae for
$\Omega_{P}(k)$ are known.  Hopkins surveys these posets, the formulae, and gives the
heuristic: \textit{Posets with order-polynomial product formulae are the same as the posets
with good dynamical behavior.}   The one poset in his list whose rowmotion dynamics were relatively
unexplored is $\VV \times [k]$, a gap this paper fills.  In separate work Hopkins and Rubey
study the dynamics of Sch\"utzenberger promotion on linear extensions of $\VV \times [k]$,
which also exhibit unusually good behavior~\cite{Hopkins-Rubey}.  

This paper is organized as follows.  In Section~\ref{sec:intro} after the introduction, we review the
toggling definition of rowmotion. Section~\ref{sec:whirling} describes whirling, and includes the equivariant
bijection which allows us to study rowmotion on $\VV_{k}$ as whirling on $k$-bounded
$P$-partitions.  Section~\ref{sec:vee} contains our main periodicity and homomesy results for rowmotion
on $\VV_{k}$, which use decompositions of the ``orbit board'' of the corresponding whirling
action into ``whorms''.  Finally, Section~\ref{sec:claw} contains the periodicity and homomesy results
which generalize to rowmotion on the ``chain of claws'' graph, $\BB_{n}\times [k]$. A
version of this paper with full proofs will appear soon on the arXiv. 

\subsection{Rowmotion as a product of toggles}\label{ss:toggles}

\begin{definition}\label{def:rowmotion}
We define the (order-ideal) rowmotion map, $\rho :\mathcal{J}(P)\to\mathcal{J}(P)$ as
follows:  For any $I\in \mathcal{J}(P)$, $\rho (I)$ is
the order ideal generated by the minimal elements of the complement of $I$, as in the example below.
\end{definition}

\begin{ex}
Here is one iteration of $\jrho$ on an order ideal with the action broken down
into its three steps: (1) complement, (2) take minimal elements, (3) saturate down. 
\[
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=white] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill=white] (4,0) circle [radius =.2];
    
\end{tikzpicture}
\xrightarrow[]{(1)}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill=white] (0,0) circle [radius =.2];
    \draw [fill=white] (1,-1) circle [radius =.2];
    \draw [fill] (1,1) circle [radius =.2];
    \draw [fill] (2,0) circle [radius =.2];
    \draw [fill] (3,1) circle [radius =.2];
    \draw [fill=white] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
    
\end{tikzpicture}
\xrightarrow[]{(2)}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill=white] (0,0) circle [radius =.2];
    \draw [fill=white] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill=white] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
    
\end{tikzpicture}
\xrightarrow[]{(3)}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill=white] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
    
\end{tikzpicture}
\]
\end{ex}

Rowmotion has an alternate definition as a composition of toggling involutions, which has
proven useful for understanding and generalizing many of its properties.  Cameron and
Fon-der-Flaass~\cite{CF95} showed that for any finite poset $P$, rowmotion can be realized
as ``toggling once at each element of $P$ along any linear extension (from top to bottom)''.
Other toggling orders also lead to interesting maps, such as Striker--Williams ``promotion''
(of order ideals) of a poset, which is toggling from left-to-right along ``files'' of a
poset~\cite{SW12}.  

\begin{definition}\label{def:toggles}
For each fixed $x\in P$ define the \emph{(order-ideal) toggle}  $\htau_x:\mathcal{J}(P)\to\mathcal{J}(P)$ by
\[
\htau_x(I) = 
\begin{cases}
I\smallsetminus \{x\}&\text{if }x\in I\text{ and }I\smallsetminus \{x\}\in \mathcal{J}(P)\\
I\cup \{x\}&\text{if }x\not\in I\text{ and }I\cup\{x\}\in\mathcal{J}(P)\\
I&\text{otherwise. }
\end{cases}
\]
\end{definition}
\noindent It is an easy exercise to show that order-ideal toggles \cite[\S 2]{CF95} are involutions,
and that toggles at incomparable elements commute (a special case of
Prop~\ref{prop:inccommute}).   

\begin{ex}
We will toggle each node down the following fixed linear extension: at each step we consider whether or not to toggle the red node in or out. 
$
\begin{tikzpicture}[scale =0.6, baseline={([yshift=-.8ex]current bounding box.center)}]

    \node at (-1,1) (2) {$2$};
    \node at (-2,2) (4) {$5$};
    \node at (0,2) (6) {$4$};
    \node at (-1,3) (12) {$7$};
    \node at (1,1) (3) {$1$};
    \node at (2,2) (9) {$3$};
    \node at (1,3) (18) {$6$};
    
    \draw[-] (2) -- (4);
    \draw[-] (2) -- (6);
    \draw[-] (3) -- (6);
    \draw[-] (3) -- (9);
    \draw[-] (4) -- (12);
    \draw[-] (6) -- (12);
    \draw[-] (6) -- (18);
    \draw[-] (9) -- (18);
\end{tikzpicture}.
$
For this linear extension we toggle the elements from top-to-bottom, then left-to-right. 
\[
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill=white] (0,0) circle [radius =.2];
    \draw [fill=] (1,-1) circle [radius =.2];
    \draw [red,fill=white] (1,1) circle [radius =.2];
    \draw [fill=] (2,0) circle [radius =.2];
    \draw [fill] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_7}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill=white] (0,0) circle [radius =.2];
    \draw [fill=] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=] (2,0) circle [radius =.2];
    \draw [red,fill] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_6}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [red,fill=white] (0,0) circle [radius =.2];
    \draw [fill=] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_5}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [red,fill] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill] (4,0) circle [radius =.2];
\end{tikzpicture}
\]
\[
\xrightarrow{\htau_4}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=white] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [red,fill] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_3}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [red,fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=white] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill] (3,-1) circle [radius =.2];
    \draw [fill=white] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_2}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=white] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [red,fill] (3,-1) circle [radius =.2];
    \draw [fill=white] (4,0) circle [radius =.2];
\end{tikzpicture}
\xrightarrow{\htau_1}
\begin{tikzpicture}[xscale = 0.75, yscale = 0.75, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0) -- (1,-1);
    \draw[-] (0,0) -- (1,1);
    \draw[-] (2,0) -- (1,-1);
    \draw[-] (2,0) -- (1,1);
    \draw[-] (2,0) -- (3,-1);
    \draw[-] (2,0) -- (3,1);
    \draw[-] (4,0) -- (3,-1);
    \draw[-] (4,0) -- (3,1);
    
    \draw [fill] (0,0) circle [radius =.2];
    \draw [fill] (1,-1) circle [radius =.2];
    \draw [fill=white] (1,1) circle [radius =.2];
    \draw [fill=white] (2,0) circle [radius =.2];
    \draw [fill=white] (3,1) circle [radius =.2];
    \draw [fill=white] (3,-1) circle [radius =.2];
    \draw [fill=white] (4,0) circle [radius =.2];
\end{tikzpicture}
\]
\end{ex}

\begin{prop}[\protect{\cite[Lemma~1]{CF95}}]
\label{prop:togall}
Let $x_1,x_2,\dots,x_p$ be any linear extension (i.e., any
order-preserving listing of the elements) of a finite poset $P$ with $p$
elements. Then the composite map $\htau_{x_1} \htau_{x_2} \cdots
\htau_{x_p}$ coincides with the rowmotion operation $\jrho$.  
\end{prop}

\section{Whirling}\label{sec:whirling}
\subsection{Whirling function between finite sets}\label{subsec:whirl}
Let $\mathcal{F}\subseteq [k]^{[n]}$ be a family of functions $f:[n]
\to [k]$. For the rest of section~\ref{subsec:whirl}, we use $\{1,\dots,k\}=[k]$ to represent the congruence
classes of $\mathbb{Z}/k\mathbb{Z}$, as opposed to the usual $\{0,1,\dots,k-1\}$. For fixed
values of $k$ and $n$, we represent such functions in \emph{one-line} notation, e.g.,  $f =
21344$ represents the function $f\in [4]^{[5]}$ with $f(1)=2$, $f(2)=1$, $f(3)=3$, $f(4)=4$,
and $f(5)=4$.  
\begin{definition}[ \protect{\cite[Definition~2.3]{JPR18}} ]
For $f\in \mathcal{F}$ we define the \emph{whirl} $w_i:\mathcal{F}\to\mathcal{F}$ at index $i$ as follows: repeatedly add 1 (modulo $k$) to the value of $f(i)$ until we get a function in $\mathcal{F}$. 
\end{definition}

\begin{wrapfigure}[12]{l}{100pt}\centering
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \tikzColoredSquare{0}{0}{4}{white!45!yellow}
    \tikzColoredSquare{1}{0}{1}{white!45!blue}
    \tikzColoredSquare{2}{0}{5}{white!45!yellow}
    
    \tikzColoredSquare{0}{-1}{6}{white!45!yellow}
    \tikzColoredSquare{1}{-1}{2}{white!45!blue}
    \tikzColoredSquare{2}{-1}{1}{white!45!red}
    
    \tikzColoredSquare{0}{-2}{3}{white!45!blue}
    \tikzColoredSquare{1}{-2}{4}{white!45!blue}
    \tikzColoredSquare{2}{-2}{2}{white!45!red}
    
    \tikzColoredSquare{0}{-3}{5}{white!45!blue}
    \tikzColoredSquare{1}{-3}{6}{white!45!blue}
    \tikzColoredSquare{2}{-3}{3}{white!45!red}
    
    \tikzColoredSquare{0}{-4}{1}{white!45!green}
    \tikzColoredSquare{1}{-4}{2}{white!45!green}
    \tikzColoredSquare{2}{-4}{4}{white!45!red}
    
    \tikzColoredSquare{0}{-5}{3}{white!45!green}
    \tikzColoredSquare{1}{-5}{5}{white!45!red}
    \tikzColoredSquare{2}{-5}{6}{white!45!red}
    
    \tikzColoredSquare{0}{-6}{4}{white!45!green}
    \tikzColoredSquare{1}{-6}{1}{white!45!orange}
    \tikzColoredSquare{2}{-6}{2}{white!45!orange}
    
    \tikzColoredSquare{0}{-7}{5}{white!45!green}
    \tikzColoredSquare{1}{-7}{3}{white!45!orange}
    \tikzColoredSquare{2}{-7}{4}{white!45!orange}
    
    \tikzColoredSquare{0}{-8}{6}{white!45!green}
    \tikzColoredSquare{1}{-8}{5}{white!45!orange}
    \tikzColoredSquare{2}{-8}{1}{white!45!yellow}
    
    \tikzColoredSquare{0}{-9}{2}{white!45!yellow}
    \tikzColoredSquare{1}{-9}{6}{white!45!orange}
    \tikzColoredSquare{2}{-9}{3}{white!45!yellow}

    
\end{tikzpicture}
\vspace{-0.5em}
\caption{}\label{fig:chunk}
\end{wrapfigure}

\begin{ex}
Let $\mathcal{F} = \{f\in [4]^{[5]}:f(1)\not =f( 2)\}$. 
If we apply $w_2$ to $f = 21344$, adding 1 in the second position gives $22344$, but this is not in $\mathcal{F}$. Adding 1 again in this position gives the result: $w_2(f)=23344$. 
\end{ex}

\noindent We will now highlight some specific results from the paper where whirling was
first introduced. Let $\mathrm{Inj}_m(n,k)$ be the set of \emph{$m$-injective functions},
that is, functions $f:[n]\to[k]$ such that $\#f^{-1}(t)\leq m$ for all $t\in[k]$. Similarly,
let $\mathrm{Sur}_m(n,k)$ be the set of \emph{$m$-surjective functions}, that is,
$f:[n]\to[k]$ such that $\#f^{-1}(t)\geq m$ for all $t\in[k]$. Note that injective functions
are $1$-injections and surjective functions are $1$-surjections. We also define the
statistic $\eta_j(f)=\# f^{-1}(\{j\})$. 

\begin{thm}\cite[Theorem~2.11]{JPR18}\label{thm:JPRthm}
Fix $\mathcal{F}$ to be either $\mathrm{Inj}_m(n,k)$ or $\mathrm{Sur}_1(n,k)$ for given $n,k,m\in \mathbb{P}$. Then under the action of $\mathbf{w}=w_n\circ w_{n-1}\circ \cdots \circ w_1$ on $\mathcal{F}$, $\eta_j$ is $\frac{n}{k}$-mesic for any $j\in[k]$ 
\end{thm}
\noindent This result is conjectured to hold for $\mathrm{Sur}_m(n,k)$, but is still open for $m>1$.
Proof details can be found in Sections 2.2--2.4 of \cite{JPR18}. 

\begin{ex}
Here is the orbit of $\mathbf{w}$ on $\mathrm{Inj}_1(3,6)$ containing $f=415$.

$$415\xrightarrow{\mathbf{w}} 621\xrightarrow{\mathbf{w}} 342\xrightarrow{\mathbf{w}} 563\xrightarrow{\mathbf{w}} 124\xrightarrow{\mathbf{w}} 356\xrightarrow{\mathbf{w}} 412\xrightarrow{\mathbf{w}} 534\xrightarrow{\mathbf{w}} 651\xrightarrow{\mathbf{w}} 263\return{\mathbf{w}}$$

Figure~\ref{fig:chunk} shows the corresponding \textit{orbit board} (a matrix whose rows are
the successive orbit elements) partitioned into chunks. Notice that each value $1,2,\dots,6$
appear exactly 5 times in this orbit of size 10, in accordance with the $1/2$-mesy of
Theorem~\ref{thm:JPRthm}. 
\end{ex}




\subsection{$k$-bounded $P$-partitions}
Now we extend the definition of whirling to $k$-bounded $P$-partitions.  
Throughout the rest of the paper, $P$ will denote a finite poset. Define $[0,k] := \{0,1,2,\dots,k\}$.

A \emph{$P$-partition} is a map $\sigma$ from $P$ to $\NN$ such that if $x<_P y$, then $\sigma(x)\geq\sigma(y)$~\cite[Ch. 3]{Sta11}.
\begin{definition}
A $k$-\emph{bounded} $P$-\emph{partition} is a function $f:P\to [0,k]$ such that if $x\leq_P y$, then  $f(x)\ge f(y)$. Let $\mathcal{F}_k(P)$ be the set of all such functions. 
\end{definition}

\noindent 
Throughout the rest of the paper we use $\{0,1,\dots,k\}$ to represent the congruence
classes of $\mathbb{Z}/(k+1)\mathbb{Z}$, as usual.
\begin{definition}
For $f\in \mathcal{F}_k(P)$ and $x\in P$, define $w_x:\mathcal{F}_k(P)\to\mathcal{F}_k(P)$, called the \emph{whirl at }$x$, as follows: repeatedly add 1 ($\textrm{mod } k+1$) to the value of $f(x)$ until we get a function in $\mathcal{F}_k(P)$.  This new function is $w_x(f)$.  \end{definition}
\noindent The case $k=1$ of the above definition recovers toggling of order ideals (Def.~\ref{def:toggles}). 
\begin{prop}\label{prop:inccommute}
If $x,y\in P$ are incomparable, then $w_xw_y(f) = w_yw_x(f)$.
\end{prop}

\begin{definition}\label{def:whirl}
Let $(x_{1},x_{2},\dots x_{p})$ be a linear extension of $P$.  Define $w:
\mathcal{F}_k(P)\rightarrow \mathcal{F}_k(P)$ by $w:=w_{x_{1}}w_{x_{2}}\dots w_{x_{p}}$.
The above proposition shows that this is well-defined, since one can get from any linear
extension to any other by a sequence of interchanges of incomparable elements.  
\end{definition}

\begin{ex}
Let $P$ be the $\VV$ poset with labels \begin{tikzpicture}[scale = .65, baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (a) {$\ell$};
    \node at (0,0) (b) {$c$};
    \node at (1,1) (c) {$r$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
\end{tikzpicture}, $k=2$, and $w=w_c w_r w_\ell$.
\[
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (a) {$0$};
    \node at (0,0) (b) {$2$};
    \node at (1,1) (c) {$2$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
\end{tikzpicture}
\xrightarrow{w_\ell}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (a) {$1$};
    \node at (0,0) (b) {$2$};
    \node at (1,1) (c) {$2$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
\end{tikzpicture}
\xrightarrow{w_r}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (a) {$1$};
    \node at (0,0) (b) {$2$};
    \node at (1,1) (c) {$0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
\end{tikzpicture}
\xrightarrow{w_c}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (a) {$1$};
    \node at (0,0) (b) {$1$};
    \node at (1,1) (c) {$0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
\end{tikzpicture}
\]
\end{ex}

There is a natural bijection between order ideals of a poset $P$ and $1$-bounded $P$-partitions in $\mathcal{F}_1(P)$. Specifically, a $1$-bounded $P$-partition in $\mathcal{F}_1(P)$ is simply the indicator function of an order ideal $I\in J(P)$. We extend this to an equivariant bijection $\mathcal{F}_k(P)\to\mathcal{J}(\mathcal{P}\times [k])$ which sends $w$ to $\jrho$,  meaning the following diagram commutes.

\[
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (0,2) (a) {$\mathcal{F}_k(P)$};
    \node at (0,0) (b) {$\mathcal{J}(P\times[k])$};
    \node at (4,2) (c) {$\mathcal{F}_k(P)$};
    \node at (4,0) (d) {$\mathcal{J}(P\times[k])$};
    \draw[<->] (a) -- (b);
    \draw[<->] (c) -- (d);
    \draw[->] (a) -- (c);
    \draw[->] (b) -- (d);
    
    \node at (2,0.25) (x) {$\jrho$};
    \node at (2,2.25) (x) {$w$};
\end{tikzpicture}
\]

We will call the chains $\{(x,1),(x,2),\dots,(x,k)\}\subseteq P\times [k]$, for $x\in P$, the \emph{fibers} of $P\times [k]$, and construct an equivariant bijection that first sends $w_x$ to order-ideal toggling down the fiber $\{(x,1),(x,2),\dots,(x,k)\}$.

\begin{lem}\label{lem:eqwhirl}
There is an equivariant bijection between $\mathcal{F}_k\left(P\right)$ and $\mathcal{J}(P\times [k])$ which sends $w_x$ to the toggle product $\tau_{(x,1)}\tau_{(x,2)}\dots\tau_{(x,k)}$.
\end{lem}

\begin{thm}\label{thm:eqwhirl}
Fix any linear extension $(x_{1},x_{2},\dots ,x_{p})\in\mathcal{L}(P)$. There is an
equivariant bijection between  $\mathcal{F}_k\left(P\right)$ and 
$\mathcal{J}(P\times [k])$
which sends whirling, $w=w_{x_{1}}w_{x_{2}}\cdots w_{x_{p}}$, to rowmotion on
$\mathcal{J}(P\times[k])$.  
\end{thm}

The following definitions will allow us to partition orbit boards of whirling into subsets called \emph{whorms}.

\begin{definition}
For any $x\in P$ and $f\in \mathcal{F}_k(P)$, define $(x,f)$ to be a \emph{whirl element}. The whirl element $(y,g)$ is \emph{whirl successive} of $(x,f)$ if either:
\begin{enumerate}
    \item $y=x$ and $g(y)=w(f)(x)=f(x)+1$, or 
    \item $x$ covers $y$, $f=g$, and $f(x)=g(y)$.
\end{enumerate}
\end{definition}
\noindent 
We consider whirl-successive elements to be whirl elements which are one step away from
each other, either by moving one covering relation down the poset or by whirling the
function at the element, and ending one label greater.  
\emph{While we must consider the entire $P$-partitions $f$ and $g$ to check whether two whirl
elements are whorm connected, we think of whirl elements as being simply $(x,f(x))$, the
location and its label, and indicate them in this way in the examples that follow.}  

\begin{definition}\label{def:whorms} Two whirl elements $(x,f)$ and $(y,g)$ are
\emph{whorm-connected} if there exists a sequence of whirl-successive elements $\{(x,f) =
(x_0,f_0), (x_1,f_1),\dots , (x_p,f_p) = (y,g)\}$. A \emph{whorm} is a maximal set of
whorm-connected whirl elements, that is, if $(x,f)$ is in a whorm and $(x,f)$ is
whorm-connected to $(y,g)$, then $(y,g)$ is in the whorm.  
\end{definition}

\begin{ex} An orbit of whirling $P$-partitions (for $P=[2]\times [2]$) with its four whorms
indicated by the same color and (redundantly) node-shape.  
\[
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    
    \draw (0,0) circle (7pt);
    \node at (0,0) (b) {$\color{black!15!blue}2$};
	\draw (-1,1) circle (7pt);
    \node at (-1,1) (a) {$\color{black!15!blue}2$};
	\draw[-] (1-0.25,1-0.25) -- (1+0.25,1-0.25) -- (1+0.25,1+0.25) -- (1-0.25,1+0.25) -- (1-0.25,1-0.25) ; %square
	\node at (1,1) (c) {$\color{black!45!purple}1$};
    \draw[-] (0,2-0.3) -- (0-0.3,2) -- (0,2+0.3) -- (0+0.3,2+0) -- (0,2-0.3) ; %diamond
    \node at (0,2) (d) {$\color{white!45!red} 0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
    \draw[-] (d) -- (a);
    \draw[-] (d) -- (c);
\end{tikzpicture}
\xrightarrow[]{w}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
	\draw[-] (0-0.25,0-0.25) -- (0+0.25,0-0.25) -- (0+0.25,0+0.25) -- (0-0.25,0+0.25) -- (0-0.25,0-0.25) ; %square
    \node at (0,0) (b) {$\color{black!45!purple}2$};
    \draw[-] (-1,1-0.3) -- (-1-0.3,1) -- (-1,1+0.3) -- (-1+0.3,1+0) -- (-1,1-0.3) ; %diamond
    \node at (-1,1) (a) {$\color{white!45!red} 1$};
	\draw[-] (1-0.25,1-0.25) -- (1+0.25,1-0.25) -- (1+0.25,1+0.25) -- (1-0.25,1+0.25) -- (1-0.25,1-0.25) ; %square    
	\node at (1,1) (c) {$\color{black!45!purple}2$};
    \draw[-] (0,2-0.3) -- (0-0.3,2) -- (0,2+0.3) -- (0+0.3,2+0) -- (0,2-0.3) ; %diamond
    \node at (0,2) (d) {$\color{white!45!red} 1$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
    \draw[-] (d) -- (a);
    \draw[-] (d) -- (c);
\end{tikzpicture}
\xrightarrow[]{w}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
	
    \draw[-] (0,0-0.3) -- (0-0.3,0) -- (0,0+0.3) -- (0+0.3,0+0) -- (0,0-0.3) ; %diamond
    \node at (0,0) (b) {$\color{white!45!red}2$};
    \draw[-] (-1,1-0.3) -- (-1-0.3,1) -- (-1,1+0.3) -- (-1+0.3,1+0) -- (-1,1-0.3) ; %diamond
    \node at (-1,1) (a) {$\color{white!45!red}2$};
    \draw[-] (1,1+0.3) -- (1-0.30,1-0.20) -- (1+0.30,1-0.20) -- (1,1+0.3) ; %triangle
    \node at (1,1) (c) {$\color{black!15!green}0$};
    \draw[-] (0,2+0.3) -- (0-0.30,2-0.20) -- (0+0.30,2-0.20) -- (0,2+0.3) ; %triangle
    \node at (0,2) (d) {$\color{black!15!green}0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
    \draw[-] (d) -- (a);
    \draw[-] (d) -- (c);
\end{tikzpicture}
\xrightarrow[]{w}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
	\draw[-] (0,0+0.3) -- (0-0.30,0-0.20) -- (0+0.30,0-0.20) -- (0,0+0.3) ; %triangle
    \node at (0,0) (b) {$\color{black!15!green}1$};
    \draw (-1,1) circle (7pt);
    \node at (-1,1) (a) {$\color{black!15!blue}0$};
	\draw[-] (1,1+0.3) -- (1-0.30,1-0.20) -- (1+0.30,1-0.20) -- (1,1+0.3) ; %triangle
    \node at (1,1) (c) {$\color{black!15!green}1$};
    \draw (0,2) circle (7pt);
    \node at (0,2) (d) {$\color{black!15!blue}0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
    \draw[-] (d) -- (a);
    \draw[-] (d) -- (c);
\end{tikzpicture}
\xrightarrow[]{w}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (0,0+0.3) -- (0-0.30,0-0.20) -- (0+0.30,0-0.20) -- (0,0+0.3) ; %triangle
    \node at (0,0) (b) {$\color{black!15!green}2$};
    \draw (-1,1) circle (7pt);
    \node at (-1,1) (a) {$\color{black!15!blue}1$};
	\draw[-] (1-0.25,1-0.25) -- (1+0.25,1-0.25) -- (1+0.25,1+0.25) -- (1-0.25,1+0.25) -- (1-0.25,1-0.25) ; %square
    \node at (1,1) (c) {$\color{black!45!purple}0$};
    \draw[-] (0-0.25,2-0.25) -- (0+0.25,2-0.25) -- (0+0.25,2+0.25) -- (0-0.25,2+0.25) -- (0-0.25,2-0.25) ; %square
    \node at (0,2) (d) {$\color{black!45!purple}0$};
    \draw[-] (a) -- (b);
    \draw[-] (c) -- (b);
    \draw[-] (d) -- (a);
    \draw[-] (d) -- (c);
\end{tikzpicture}\return{w}
\]

\end{ex}

\section{Periodicity and homomesy for rowmotion on $\VV\times[k]$}\label{sec:vee}
In this section we consider the dynamics of rowmotion acting on the order ideals of the chain of V's poset
$\VV_{k}$,  establishing its periodicity and finding interesting examples of homomesy. 


\begin{wrapfigure}[12]{l}{80pt}
    \begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \node at (-1,1) (l2) {$\ell_1$};
    \node at (0,0) (c3) {$c_1$};
    \node at (1,1) (r2) {$r_1$};
    \draw[-] (c3) -- (l2);
    \draw[-] (c3) -- (r2);
    \node at (-1,2) (l3) {$\ell_2$};
    \node at (0,1) (c4) {$c_2$};
    \node at (1,2) (r3) {$r_2$};
    \draw[-] (c4) -- (l3);
    \draw[-] (c4) -- (r3);
    \node at (-1,3) (l4) {$\ell_3$};
    \node at (0,2) (c5) {$c_3$};
    \node at (1,3) (r4) {$r_3$};
    \draw[-] (c5) -- (l4);
    \draw[-] (c5) -- (r4);
    \node at (-1,5) (l5) {$\ell_k$};
    \node at (0,4) (c6) {$c_k$};
    \node at (1,5) (r5) {$r_k$};
    \draw[-] (c6) -- (l5);
    \draw[-] (c6) -- (r5);
    \node at (-1,4) (l) {$\vdots$};
    \node at (0,3) (c) {$\vdots$};
    \node at (1,4) (r) {$\vdots$};
    
    \draw[-] (c3) -- (c4);
    \draw[-] (c4) -- (c5);
    \draw[-] (c5) -- (c);
    \draw[-] (c) -- (c6);
    
    \draw[-] (l2) -- (l3);
    \draw[-] (l3) -- (l4);
    \draw[-] (l4) -- (l);
    \draw[-] (l) -- (l5);
    
    \draw[-] (r2) -- (r3);
    \draw[-] (r3) -- (r4);
    \draw[-] (r4) -- (r);
    \draw[-] (r) -- (r5);
\end{tikzpicture}
\caption{}\label{fig:vee}
\end{wrapfigure}

\begin{definition}
Let $\VV $ be the 3-element poset with Hasse diagram
$
\begin{tikzpicture}[scale = 0.5, baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (0,0);
    \draw[-] (1,1) -- (0,0);
    
    \draw [fill] (0,0) circle [radius =.1];
    \draw [fill] (-1,1) circle [radius =.1];
    \draw [fill] (1,1) circle [radius =.1];
\end{tikzpicture}
$,
and define $\VV_k=\VV\times[k]$, where $[k]$ is the chain poset.  We call $\VV_{k}$ the
\emph{chain of V's poset}.  
\end{definition}
\begin{ex}
Figure~\ref{fig:vee} shows the Hasse diagram of $\VV_k$ with our vertex-labeling convention.
\end{ex}

Our main goals for this section are the following theorems.  We will leverage the
equivariant bijection and the notion of whorms from the last section.  

\begin{thm}\label{thm:Vorder}
The order of rowmotion on $\mathcal{J}(\VV_k)$ is $2(k+2)$.
\end{thm}


\begin{thm}\label{thm:Vhom}
Let $\chi_s$ be the indicator function for $s\in \VV_k$. We have the following homomesies for the action of $\jrho$ on $\mathcal{J}(\VV_k)$
\begin{enumerate}
    \item The statistic $\chi_{\ell_i}-\chi_{r_i}$ is $0$-mesic for all $i\in [k]$.
    \item The statistic $\chi_{\ell_1} + \chi_{r_1} - \chi_{c_k}$ is $\frac{2(k-1)}{k+2}$-mesic.
\end{enumerate}

\end{thm}



\begin{ex} \label{exorbit} This $\jrho$-orbit on $\mathcal{J}(\VV_4)$ has size 4, which
divides $2(4+2)=12$. The homomesies are also easily checked, e.g., across the orbit the total number of
elements at rank 1 in the side fibers is 6, minus the two at the top of the center fiber,
for an average of $\frac{6-2}{4}=1 = \frac{2(4-1)}{4+2}$, agreeing with
Theorem~\ref{thm:Vhom}(2).   
\[
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (-1,4);
    \draw[-] (1,1) -- (1,4);
    \draw[-] (0,0) -- (0,3);

    \draw[-] (-1,1) -- (0,0);
    \draw[-] (1,1) -- (0,0);
    
    \draw [fill] (-1,1) circle [radius =.1];
    \draw [fill] (0,0) circle [radius =.1];
    \draw [fill] (1,1) circle [radius =.1];
    
    \draw[-] (-1,2) -- (0,1);
    \draw[-] (1,2) -- (0,1);
    
    \draw [fill=white] (-1,2) circle [radius =.1];
    \draw [fill] (0,1) circle [radius =.1];
    \draw [fill] (1,2) circle [radius =.1];
    
    \draw[-] (-1,3) -- (0,2);
    \draw[-] (1,3) -- (0,2);
    
    \draw [fill=white] (-1,3) circle [radius =.1];
    \draw [fill] (0,2) circle [radius =.1];
    \draw [fill] (1,3) circle [radius =.1];
    
    \draw[-] (-1,4) -- (0,3);
    \draw[-] (1,4) -- (0,3);
    
    \draw [fill=white] (-1,4) circle [radius =.1];
    \draw [fill=white] (0,3) circle [radius =.1];
    \draw [fill=white] (1,4) circle [radius =.1];
\end{tikzpicture}
\xrightarrow{\jrho}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (-1,4);
    \draw[-] (1,1) -- (1,4);
    \draw[-] (0,0) -- (0,3);

    \draw[-] (-1,1) -- (0,0);
    \draw[-] (1,1) -- (0,0);
    
    \draw [fill] (-1,1) circle [radius =.1];
    \draw [fill] (0,0) circle [radius =.1];
    \draw [fill=white] (1,1) circle [radius =.1];
    
    \draw[-] (-1,2) -- (0,1);
    \draw[-] (1,2) -- (0,1);
    
    \draw [fill] (-1,2) circle [radius =.1];
    \draw [fill] (0,1) circle [radius =.1];
    \draw [fill=white] (1,2) circle [radius =.1];
    
    \draw[-] (-1,3) -- (0,2);
    \draw[-] (1,3) -- (0,2);
    
    \draw [fill=white] (-1,3) circle [radius =.1];
    \draw [fill] (0,2) circle [radius =.1];
    \draw [fill=white] (1,3) circle [radius =.1];
    
    \draw[-] (-1,4) -- (0,3);
    \draw[-] (1,4) -- (0,3);
    
    \draw [fill=white] (-1,4) circle [radius =.1];
    \draw [fill] (0,3) circle [radius =.1];
    \draw [fill=white] (1,4) circle [radius =.1];
\end{tikzpicture}
\xrightarrow{\jrho}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (-1,4);
    \draw[-] (1,1) -- (1,4);
    \draw[-] (0,0) -- (0,3);

    \draw[-] (-1,1) -- (0,0);
    \draw[-] (1,1) -- (0,0);
    
    \draw [fill] (-1,1) circle [radius =.1];
    \draw [fill] (0,0) circle [radius =.1];
    \draw [fill] (1,1) circle [radius =.1];
    
    \draw[-] (-1,2) -- (0,1);
    \draw[-] (1,2) -- (0,1);
    
    \draw [fill] (-1,2) circle [radius =.1];
    \draw [fill] (0,1) circle [radius =.1];
    \draw [fill=white] (1,2) circle [radius =.1];
    
    \draw[-] (-1,3) -- (0,2);
    \draw[-] (1,3) -- (0,2);
    
    \draw [fill] (-1,3) circle [radius =.1];
    \draw [fill] (0,2) circle [radius =.1];
    \draw [fill=white] (1,3) circle [radius =.1];
    
    \draw[-] (-1,4) -- (0,3);
    \draw[-] (1,4) -- (0,3);
    
    \draw [fill=white] (-1,4) circle [radius =.1];
    \draw [fill=white] (0,3) circle [radius =.1];
    \draw [fill=white] (1,4) circle [radius =.1];
\end{tikzpicture}
\xrightarrow{\jrho}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (-1,4);
    \draw[-] (1,1) -- (1,4);
    \draw[-] (0,0) -- (0,3);

    \draw[-] (-1,1) -- (0,0);
    \draw[-] (1,1) -- (0,0);
    
    \draw [fill=white] (-1,1) circle [radius =.1];
    \draw [fill] (0,0) circle [radius =.1];
    \draw [fill] (1,1) circle [radius =.1];
    
    \draw[-] (-1,2) -- (0,1);
    \draw[-] (1,2) -- (0,1);
    
    \draw [fill=white] (-1,2) circle [radius =.1];
    \draw [fill] (0,1) circle [radius =.1];
    \draw [fill] (1,2) circle [radius =.1];
    
    \draw[-] (-1,3) -- (0,2);
    \draw[-] (1,3) -- (0,2);
    
    \draw [fill=white] (-1,3) circle [radius =.1];
    \draw [fill] (0,2) circle [radius =.1];
    \draw [fill=white] (1,3) circle [radius =.1];
    
    \draw[-] (-1,4) -- (0,3);
    \draw[-] (1,4) -- (0,3);
    
    \draw [fill=white] (-1,4) circle [radius =.1];
    \draw [fill] (0,3) circle [radius =.1];
    \draw [fill=white] (1,4) circle [radius =.1];
\end{tikzpicture}\return{\jrho}
\]
\end{ex}

To prove these theorems we utilize our equivariant bijection (Theorem~\ref{thm:eqwhirl}) from $\mathcal{J}(\VV_k)$ 
to $\mathcal{F}_k(\VV)$, 
then represent the latter by triples $f=(\ell,c,r)$ with $\ell \leq  c$ and $r\leq c$.  This bijection $\phi$ sends an order ideal $I$ to a triple $(\ell,c,r)$, counting the number
of elements of the order ideal in the left, center, and right fibers respectively.  

\begin{ex} Here is the orbit of $\mathcal{F}_4(\VV)$ corresponding
to Example~\ref{exorbit}.  
\[(1,3,3)\xrightarrow{w} (2,4,0)\xrightarrow{w}(3,3,1)\xrightarrow{w} (0,4,2) \return{w} \]
\end{ex}


\begin{prop}
The number of order ideals of $\VV_k$ is given by 
$|\mathcal{J}(\VV_k)| = \frac{k(k+1)(2k+1)}{6}$. 
\end{prop}

\begin{wrapfigure}[14]{l}{130pt}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \tikzColoredSquare{0}{0}{1}{white!45!blue}
    \tikzColoredSquare{0}{-1}{2}{white!45!blue}
    \tikzColoredSquare{0}{-2}{3}{white!45!blue}
    \tikzColoredSquare{0}{-3}{4}{white!45!blue}
    \tikzColoredSquare{1}{-3}{4}{white!45!blue}
    \tikzColoredSquare{2}{0}{0}{white!45!green}
    \tikzColoredSquare{2}{-1}{1}{white!45!green}
    \tikzColoredSquare{2}{-2}{2}{white!45!green}
    \tikzColoredSquare{2}{-3}{3}{white!45!green}
    \tikzColoredSquare{2}{-4}{4}{white!45!green}
    \tikzColoredSquare{1}{-4}{4}{white!45!green}
    \tikzColoredSquare{0}{-4}{0}{white!45!red}
    \tikzColoredSquare{0}{-5}{1}{white!45!red}
    \tikzColoredSquare{1}{-5}{1}{white!45!red}
    \tikzColoredSquare{1}{-6}{2}{white!45!red}
    \tikzColoredSquare{1}{-7}{3}{white!45!red}
    \tikzColoredSquare{1}{-8}{4}{white!45!red}
    \tikzColoredSquare{2}{-5}{0}{white!45!yellow}
    \tikzColoredSquare{2}{-6}{1}{white!45!yellow}
    \tikzColoredSquare{2}{-7}{2}{white!45!yellow}
    \tikzColoredSquare{2}{-8}{3}{white!45!yellow}
    \tikzColoredSquare{2}{-9}{4}{white!45!yellow}
    \tikzColoredSquare{1}{-9}{4}{white!45!yellow}
    \tikzColoredSquare{0}{-6}{0}{white!45!orange}
    \tikzColoredSquare{0}{-7}{1}{white!45!orange}
    \tikzColoredSquare{0}{-8}{2}{white!45!orange}
    \tikzColoredSquare{0}{-9}{3}{white!45!orange}
    \tikzColoredSquare{0}{-10}{4}{white!45!orange}
    \tikzColoredSquare{1}{-10}{4}{white!45!orange}
    \tikzColoredSquare{2}{-10}{0}{white!75!teal}
    \tikzColoredSquare{2}{-11}{1}{white!75!teal}
    \tikzColoredSquare{1}{-11}{1}{white!75!teal}
    \tikzColoredSquare{1}{0}{2}{white!75!teal}
    \tikzColoredSquare{1}{-1}{3}{white!75!teal}
    \tikzColoredSquare{1}{-2}{4}{white!75!teal}
    \tikzColoredSquare{0}{-11}{0}{white!45!blue}
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]

    \def\m{0}
    \tikzColoredSquare{0}{\m}{0}{white!45!blue}
    \tikzColoredSquare{1}{\m}{2}{white!45!red}
    \tikzColoredSquare{2}{\m}{0}{white!45!blue}
    
    \def\m{-1}
    \tikzColoredSquare{0}{\m}{1}{white!45!blue}
    \tikzColoredSquare{1}{\m}{3}{white!45!red}
    \tikzColoredSquare{2}{\m}{1}{white!45!blue}
    
    \def\m{-2}
    \tikzColoredSquare{0}{\m}{2}{white!45!blue}
    \tikzColoredSquare{1}{\m}{4}{white!45!red}
    \tikzColoredSquare{2}{\m}{2}{white!45!blue}
    
    \def\m{-3}
    \tikzColoredSquare{0}{\m}{3}{white!45!blue}
    \tikzColoredSquare{1}{\m}{3}{white!45!blue}
    \tikzColoredSquare{2}{\m}{3}{white!45!blue}
    
    \def\m{-4}
    \tikzColoredSquare{0}{\m}{0}{white!45!red}
    \tikzColoredSquare{1}{\m}{4}{white!45!blue}
    \tikzColoredSquare{2}{\m}{0}{white!45!red}
    
    \def\m{-5}
    \tikzColoredSquare{0}{\m}{1}{white!45!red}
    \tikzColoredSquare{1}{\m}{4}{white!45!blue}
    \tikzColoredSquare{2}{\m}{1}{white!45!red}
    
    \def\m{-5}
    \tikzColoredSquare{0}{\m}{1}{white!45!red}
    \tikzColoredSquare{1}{\m}{1}{white!45!red}
    \tikzColoredSquare{2}{\m}{1}{white!45!red}
\end{tikzpicture}
\caption{}\label{fig:veetwoorbits}
\end{wrapfigure}

\subsection{Center-seeking whorms}
To show that the order of $\jrho$ on $\mathcal{J}({\VV }_k)$ is $2(k+2)$ we end up proving
something stronger, namely that $\jrho^{k+2}(I)$ is the reflection of $I$ across the the
center chain. Our method is to investigate the whorms that arise from repeatedly
whirling a $k$-bounded $P$-partition. 

Recall from Definition~\ref{def:whorms} that, given a whirling orbit board,
$\mathcal{O}=\{f,w(f), w^2(f),\dots\}$ of $w$ on $\mathcal{F}_k(\VV)$, a whorm $\varsigma$ is a
maximal set of whorm-connected elements. Figure~\ref{fig:veetwoorbits} shows two orbit boards
of $\mathcal{F}_4(\VV)$, one with six whorms and one with two whorms. Notice that each
whorm in the second orbit has two ``starting" positions.  


\begin{wrapfigure}[14]{R}{125pt}

\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \tikzColoredSquare{0}{0}{1}{white}
    \tikzColoredSquare{0}{-1}{2}{white}
    \tikzColoredSquare{0}{-2}{3}{white}
    \tikzColoredSquare{0}{-3}{4}{white}
    \tikzColoredSquare{1}{-3}{4}{white}
    \tikzColoredSquare{2}{-1}{0}{white}
    \tikzColoredSquare{2}{-2}{1}{white}
    \tikzColoredSquare{2}{-3}{2}{white}
    \tikzColoredSquare{2}{-4}{3}{white}
    \tikzColoredSquare{1}{-4}{3}{white}
    \tikzColoredSquare{1}{-5}{4}{white}
    \tikzColoredSquarech{0}{-4}{0}{white!45!red}
    \tikzColoredSquarech{0}{-5}{1}{white!45!red}
    \tikzColoredSquarech{0}{-6}{2}{white!45!red}
    \tikzColoredSquarech{1}{-6}{2}{white!45!red}
    \tikzColoredSquarech{1}{-7}{3}{white!45!red}
    \tikzColoredSquarech{1}{-8}{4}{white!45!red}
    \tikzColoredSquare{2}{-5}{0}{white}
    \tikzColoredSquare{2}{-6}{1}{white}
    \tikzColoredSquare{2}{-7}{2}{white}
    \tikzColoredSquare{2}{-8}{3}{white}
    \tikzColoredSquare{2}{-9}{4}{white}
    \tikzColoredSquare{1}{-9}{4}{white}
    \tikzColoredSquare{0}{-7}{0}{white}
    \tikzColoredSquare{0}{-8}{1}{white}
    \tikzColoredSquare{0}{-9}{2}{white}
    \tikzColoredSquare{0}{-10}{3}{white}
    \tikzColoredSquare{1}{-10}{3}{white}
    \tikzColoredSquare{1}{-11}{4}{white}
    \tikzColoredSquare{2}{-10}{0}{white}
    \tikzColoredSquare{2}{-11}{1}{white}
    \tikzColoredSquare{2}{0}{2}{white}
    \tikzColoredSquare{1}{0}{2}{white}
    \tikzColoredSquare{1}{-1}{3}{white}
    \tikzColoredSquare{1}{-2}{4}{white}
    \tikzColoredSquare{0}{-11}{0}{white}
\end{tikzpicture}\hspace{1cm}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \tikzColoredSquare{0}{0}{1}{white!45!yellow}
    \tikzColoredSquare{0}{-1}{2}{white!45!yellow}
    \tikzColoredSquare{0}{-2}{3}{white!45!yellow}
    \tikzColoredSquare{0}{-3}{4}{white!45!yellow}
    \tikzColoredSquare{1}{-3}{4}{white!45!yellow}
    \tikzColoredSquarenwl{2}{-1}{0}{white!45!green}
    \tikzColoredSquarenwl{2}{-2}{1}{white!45!green}
    \tikzColoredSquarenwl{2}{-3}{2}{white!45!green}
    \tikzColoredSquarenwl{2}{-4}{3}{white!45!green}
    \tikzColoredSquarenwl{1}{-4}{3}{white!45!green}
    \tikzColoredSquarenwl{1}{-5}{4}{white!45!green}
    \tikzColoredSquarech{0}{-4}{0}{white!45!red}
    \tikzColoredSquarech{0}{-5}{1}{white!45!red}
    \tikzColoredSquarech{0}{-6}{2}{white!45!red}
    \tikzColoredSquarech{1}{-6}{2}{white!45!red}
    \tikzColoredSquarech{1}{-7}{3}{white!45!red}
    \tikzColoredSquarech{1}{-8}{4}{white!45!red}
    \tikzColoredSquarehl{2}{-5}{0}{white!45!blue}
    \tikzColoredSquarehl{2}{-6}{1}{white!45!blue}
    \tikzColoredSquarehl{2}{-7}{2}{white!45!blue}
    \tikzColoredSquarehl{2}{-8}{3}{white!45!blue}
    \tikzColoredSquarehl{2}{-9}{4}{white!45!blue}
    \tikzColoredSquarehl{1}{-9}{4}{white!45!blue}
    \tikzColoredSquare{0}{-7}{0}{white!45!orange}
    \tikzColoredSquare{0}{-8}{1}{white!45!orange}
    \tikzColoredSquare{0}{-9}{2}{white!45!orange}
    \tikzColoredSquare{0}{-10}{3}{white!45!orange}
    \tikzColoredSquare{1}{-10}{3}{white!45!orange}
    \tikzColoredSquare{1}{-11}{4}{white!45!orange}
    \tikzColoredSquare{2}{-10}{0}{white!75!teal}
    \tikzColoredSquare{2}{-11}{1}{white!75!teal}
    \tikzColoredSquare{2}{0}{2}{white!75!teal}
    \tikzColoredSquare{1}{0}{2}{white!75!teal}
    \tikzColoredSquare{1}{-1}{3}{white!75!teal}
    \tikzColoredSquare{1}{-2}{4}{white!75!teal}
    \tikzColoredSquare{0}{-11}{0}{white!45!yellow}
\end{tikzpicture}
    \caption{}\label{fig:onewhorm}
\end{wrapfigure}

Each whorm in an orbit board of $\VV\times [k]$ starts on the left, or the right, or both left and
right; we call the former \emph{one-tailed} and the later \emph{two-tailed}. 
Since these whorms move down the orbit board at every step, except for one move to the center, we consider
them as  a sequence of function values in
the orbit board which start at $0$ and end at $k$, where one value is repeated when moving
into the center.  We call these \emph{center-seeking whorms}.
(Since an orbit board is actually a cylinder, we have a ``can of worms'' to deal with.)
In the left orbit of  Figure~\ref{fig:onewhorm} we isolate one
example of a left whorm: 
$\varsigma = \{(\ell,(0,3,3)),(\ell,(1,4,0)),(\ell,(2,2,1)),(c,(2,2,1)), (c,(0,3,2)), (c,(1,4,3))\}$,
visualized within an orbit board of $\mathcal{F}_4(\VV)$.  It is easy to see that an orbit
board is tiled either entirely by one-tailed whorms or entirely by two-tailed whorms.  (See
the discussion at the start of Section~\ref{sec:claw}.) 

We first observe that all whorms have $k+2$ elements, since each contains the $k+1$
elements $0,...,k$, exactly one of which is doubled. 

Define $\oc(\varsigma): = 1+\min \{f(c): (c,f)\in \varsigma\}$, the number of elements in the
outer columns and $\cc(\varsigma): = k+2 - \oc(\varsigma)$, the number of elements in the center column. 
For the red whorm in the orbit on the left of Figure~\ref{fig:onewhorm}, $\oc(\varsigma) = 3$ and $\cc(\varsigma) = 3$. 

\begin{ex} \label{ex:bp}
The right orbit board in Figure~\ref{fig:onewhorm} is the previous example with all the whorms
colored. The number of elements in the left column of the  yellow, red, and
orange whorms are $5,3,4$ respectively,  and the orbit board is of length $12$. 
\end{ex}


It follows that the order of whirling divides the sum of  $\oc(\varsigma)$ over all whorms
$\varsigma\in S$. In the setting of $\mathcal{F}_k(\VV)$, as long as we know
$\oc(\varsigma)$ and whether $f(\ell) = 0$, $f(r) =0$, or both, then we can recover the
entire whorm.  

\begin{definition}
We place a circular order on the whorms. Let $\varsigma_1$ and $\varsigma_2$ be whorms
in an orbit board of $\mathcal{F}_k(\VV)$. If there exists $(c, f) \in \varsigma_1$ with
$f(c) = k$ such that $(c, w(f))\in \varsigma_2$, then we say $\varsigma_2$ is \emph{in front of}
$\varsigma_1$. We call a sequence of whorms \emph{consecutive} if each is in front of the next. In a one-tailed orbit board, consecutive whorms alternate starting from the left
and right.  
\end{definition} 

\begin{ex}
In Figure~\ref{fig:onewhorm} the blue (horizontal lines) whorm is in front of the red
(crosshatch) whorm,
which is in front of the green (northwest lines) whorm.  
\end{ex}

\begin{lem}\label{lem:bp_consecutive}
Assume an orbit board $\mathcal{O}$ of $w$ on $\mathcal{F}_k(\VV)$ has all one-tailed
whorms. Let $\varsigma_1,\varsigma_2,$ and $\varsigma_3$ be three consecutive whorms, that
is, $\varsigma_3$ is in front of $\varsigma_2$ which is in front of $\varsigma_1$ in
$\mathcal{O}$. Then, $\oc(\varsigma_1)+\oc(\varsigma_2)+\oc(\varsigma_3)=2(k+2).$ Otherwise, if there are two-tailed whorms, then $\oc(\varsigma_1)+\oc(\varsigma_2) = k+2$.
\end{lem}

\noindent In fact, the entire orbit board can be reconstructed simply from knowing the
values of $\oc(\varsigma_1)$ and $\oc(\varsigma_2)$ for two consecutive
whorms in the one-tailed case, and from a single $\oc(\varsigma_1)$ in the two-tailed case.  

\begin{ex}
In Figure~\ref{fig:onewhorm} we have $k=4$, $\oc(\text{green})=4$, $\oc(\text{red})=3 $, and $\oc(\text{blue})=5$, which sum to $12=2(4+2)$.  
\end{ex}

\begin{lem}\label{lem:srotate}
Given an orbit board with one-tailed whorms, let $\varsigma_1$, $\varsigma_2$, $\varsigma_3,\varsigma_4$ be consecutive, then $\oc(\varsigma_4) = \oc(\varsigma_1).$ Furthermore, if the orbit board contains two-tailed whorms, then $\oc(\varsigma_1) = \oc(\varsigma_3)$.
\end{lem}


\begin{wrapfigure}[11]{r}{75pt}  
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1.5) -- (-1,3.5);
    \draw[-] (1,1.5) -- (1,3.5);
    \draw[-] (0,0.5) -- (0,2.5);

    \node at (-1,1) {$\vdots$};
    \node at (0,0) {$\vdots$};
    \node at (1,1) {$\vdots$};
    
    \draw[-] (-1,2) -- (0,1);
    \draw[-] (1,2) -- (0,1);
    
    \draw [fill=white] (-1,2) circle [radius =.1];
    \draw [fill] (0,1) circle [radius =.1];
    \draw [fill=white] (1,2) circle [radius =.1];
    
    \draw[-] (-1,3) -- (0,2);
    \draw[-] (1,3) -- (0,2);
    
    \draw [fill] (-1,3) circle [radius =.1];
    \draw [fill=white] (0,2) circle [radius =.1];
    \draw [fill] (1,3) circle [radius =.1];
    
    \node at (-1,4) {$\vdots$};
    \node at (0,3) {$\vdots$};
    \node at (1,4)  {$\vdots$};
\end{tikzpicture}
\caption{}\label{fig:flux}
\end{wrapfigure}
Notice that for orbits with one-tailed whorms, we are \emph{not} claiming the board starts to
repeat; since  whorms alternate sides, $\varsigma_4$ will start on the opposite side from 
$\varsigma_1$. If we keep applying the previous Lemma to even more consecutive whorms, we
see $\oc(\varsigma_5)=\oc(\varsigma_2)$ and $\oc(\varsigma_6) = \oc(\varsigma_3)$. Finally
we get $\oc(\varsigma_7)=\oc(\varsigma_1)$ and the pattern repeats. Therefore there are at
most six unique whorms in a one-tailed orbit board.
\begin{lem} \label{lem:sixwhorms}
Given an orbit board with one-tailed whorms, there are at most six distinct whorms.
\end{lem}


\begin{thm}
Let $(x,y,z)\in \mathcal{F}_k(\VV)$, then $w^{k+2}(x,y,z) = (z,y,x)$.
\end{thm}

\begin{cor}\label{thm:veeorder}
The order of $w$ on $\mathcal{F}_k(\VV)$ divides $2(k+2)$.
\end{cor}


\begin{lem}\label{lem:fluxpairs}
 Under the action of rowmotion on order ideals of
$\mathcal{J}(\VV_k)$, the difference of successive flux-capacitor indicator functions, $F_{i} - F_{i+1}$ is
$\frac{3}{k+2}$-mesic for $i\in [2,k-1]$.
\end{lem}

This lemma can be generalized to the following theorem. 

\begin{thm}\label{thm:flux}
For $k>1$. Let $F_i = \chi_{\ell_i}+\chi_{r_i} + \chi_{c_{i-1}}$.  Under the action of
rowmotion on order ideals of $\mathcal{J}(\VV_k)$, the difference of arbitrary flux-
capacitors is $F_{i} - F_{j}$ is $\frac{3(j-i)}{k+2}$-mesic.
\end{thm}


\section{Periodicity and homomesy for rowmotion on $\BB_n \times [k]$} \label{sec:claw}
We define the \emph{claw poset} $\BB_n = \{ b_1,\dots,b_n,\widehat{0}\}$ where each $b_{i}$ covers $\widehat{0}$.  For example, the Hasse diagram of $\BB_4$ would be \begin{tikzpicture}[scale=0.6,baseline={([yshift=-.8ex]current bounding box.center)}]
    \draw[-] (-1,1) -- (1.66,0);
    \draw[-] (-.33,1) -- (1.66,0);
    \draw[-] (.33,1) -- (1.66,0);
    \draw[-] (1,1) -- (1.66,0);
    
    \draw [fill] (1.66,0) circle [radius =.1];
    \draw [fill] (-1,1) circle [radius =.1];
    \draw [fill] (-.33,1) circle [radius =.1];
    \draw [fill] (1,1) circle [radius =.1];
    \draw [fill] (.33,1) circle [radius =.1];
\end{tikzpicture}.




Using the established equivariant bijection between $\mathcal{J}(\BB_n\times [k])$ and
$k$-bounded $P$-partitions $\mathcal{F}_k(\BB_n)$ that sends rowmotion to whirling,
we can prove similar homomesies and periodicity to that of $\BB_2=\VV$. Now instead of
\emph{triples} of numbers, we will consider orbit boards of \emph{$(n+1)$-tuples} on $[0,k]$, 
$\left(f(b_{1}),f(b_{2}),\dots ,f(b_{n}),f(\widehat{0})\right)$, satisfying $f(b_{i})\leq
f(\widehat{0})$ for each $i\in [n]$.  

\begin{wrapfigure}[16]{l}{85pt}
    
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)},scale=0.9]
    \tikzColoredSquare{4}{0}{3}{magenta!100}
    
    \tikzColoredSquare{0}{0}{1}{green!100}
    \tikzColoredSquare{1}{0}{0}{orange!100}
    \tikzColoredSquare{2}{0}{1}{green!100}
    \tikzColoredSquare{3}{0}{2}{lime!100}
    
    \tikzColoredSquare{4}{-1}{3}{lime!100}
    
    \tikzColoredSquare{0}{-1}{2}{green!100}
    \tikzColoredSquare{1}{-1}{1}{orange!100}
    \tikzColoredSquare{2}{-1}{2}{green!100}
    \tikzColoredSquare{3}{-1}{3}{lime!100}

    \tikzColoredSquare{4}{-2}{3}{green!100}
    
    \tikzColoredSquare{0}{-2}{3}{green!100}
    \tikzColoredSquare{1}{-2}{2}{orange!100}
    \tikzColoredSquare{2}{-2}{3}{green!100}
    \tikzColoredSquare{3}{-2}{0}{gray!100}
    
    \tikzColoredSquare{4}{-3}{3}{orange!100}
    
    \tikzColoredSquare{0}{-3}{0}{blue!100}
    \tikzColoredSquare{1}{-3}{3}{orange!100}
    \tikzColoredSquare{2}{-3}{0}{blue!100}
    \tikzColoredSquare{3}{-3}{1}{gray!100}
    
    \tikzColoredSquare{4}{-4}{2}{gray!100}
    
    \tikzColoredSquare{0}{-4}{1}{blue!100}
    \tikzColoredSquare{1}{-4}{0}{purple!100}
    \tikzColoredSquare{2}{-4}{1}{blue!100}
    \tikzColoredSquare{3}{-4}{2}{gray!100}
    
    \tikzColoredSquare{4}{-5}{3}{gray!100}
    
    \tikzColoredSquare{0}{-5}{2}{blue!100}
    \tikzColoredSquare{1}{-5}{1}{purple!100}
    \tikzColoredSquare{2}{-5}{2}{blue!100}
    \tikzColoredSquare{3}{-5}{0}{blue!50!white}
    
    \tikzColoredSquare{4}{-6}{3}{blue!100}
    
    \tikzColoredSquare{0}{-6}{3}{blue!100}
    \tikzColoredSquare{1}{-6}{2}{purple!100}
    \tikzColoredSquare{2}{-6}{3}{blue!100}
    \tikzColoredSquare{3}{-6}{1}{blue!50!white}
    
    \tikzColoredSquare{4}{-7}{3}{purple!100}
    
    \tikzColoredSquare{0}{-7}{0}{red!100}
    \tikzColoredSquare{1}{-7}{3}{purple!100}
    \tikzColoredSquare{2}{-7}{0}{red!100}
    \tikzColoredSquare{3}{-7}{2}{blue!50!white}
    
    \tikzColoredSquare{4}{-8}{3}{blue!50!white}
    
    \tikzColoredSquare{0}{-8}{1}{red!100}
    \tikzColoredSquare{1}{-8}{0}{pink!100}
    \tikzColoredSquare{2}{-8}{1}{red!100}
    \tikzColoredSquare{3}{-8}{3}{blue!50!white}
    
    \tikzColoredSquare{4}{-9}{2}{red!100}
    
    \tikzColoredSquare{0}{-9}{2}{red!100}
    \tikzColoredSquare{1}{-9}{1}{pink!100}
    \tikzColoredSquare{2}{-9}{2}{red!100}
    \tikzColoredSquare{3}{-9}{0}{gray!50}
    
    \tikzColoredSquare{4}{-10}{3}{red!100}
    
    \tikzColoredSquare{0}{-10}{0}{yellow!100}
    \tikzColoredSquare{1}{-10}{2}{pink!100}
    \tikzColoredSquare{2}{-10}{0}{yellow!100}
    \tikzColoredSquare{3}{-10}{1}{gray!50}
    
    \tikzColoredSquare{4}{-11}{3}{pink!100}
    
    \tikzColoredSquare{0}{-11}{1}{yellow!100}
    \tikzColoredSquare{1}{-11}{3}{pink!100}
    \tikzColoredSquare{2}{-11}{1}{yellow!100}
    \tikzColoredSquare{3}{-11}{2}{gray!50}
    
    \tikzColoredSquare{4}{-12}{3}{gray!50}
    
    \tikzColoredSquare{0}{-12}{2}{yellow!100}
    \tikzColoredSquare{1}{-12}{0}{magenta!100}
    \tikzColoredSquare{2}{-12}{2}{yellow!100}
    \tikzColoredSquare{3}{-12}{3}{gray!50}
    
    \tikzColoredSquare{4}{-13}{3}{yellow!100}
    
    \tikzColoredSquare{0}{-13}{3}{yellow!100}
    \tikzColoredSquare{1}{-13}{1}{magenta!100}
    \tikzColoredSquare{2}{-13}{3}{yellow!100}
    \tikzColoredSquare{3}{-13}{0}{lime!100}
    
    \tikzColoredSquare{4}{-14}{2}{magenta!100}
    
    \tikzColoredSquare{0}{-14}{0}{green!100}
    \tikzColoredSquare{1}{-14}{2}{magenta!100}
    \tikzColoredSquare{2}{-14}{0}{green!100}
    \tikzColoredSquare{3}{-14}{1}{lime!100}

\end{tikzpicture}
\caption{}\label{fig:Borbit}
\end{wrapfigure}

In Figure~\ref{fig:Borbit}, note that if two entries are the same among the first $n$ in a given row, then
those positions (columns) remain the same throughout the \emph{entire} orbit board.  This is
because the entries $b_1,\dots, b_n$ represent the result of whirling at \emph{incomporable}
elements of the poset $\BB_n$. Furthermore, these two entries must belong to the same whorm,
because each will be whorm-connected via $\widehat{0}$ exactly when their value matches the
value of the last entry. These observations will allow us to generalize our peridocity and homomesy
results from $\VV$ to $\BB_n$. 

\begin{definition}
For $A \subseteq [0,k]$, define the family of order-reversing maps $\mathcal{F}_k^{A}(\BB_n) = \{f: f\in\mathcal{F}_k(\BB_n)\text{ and }f(b_j) \in A\text{ for all } j \in[n]\}$. For any fixed $A$ we denote $\overline{w}$ to be whirling on the non-$\widehat{0}$  elements of order-reversing maps $f \in \mathcal{F}_k^{A}(\BB_n)$. Which is equivalent to incrementing each non-$\widehat{0}$ value,  but only allowing values within $A$.

\end{definition}
Given $f\in \mathcal{F}_k(\BB_n)$, set $A(f) = \{a: f(b_j) = a \text{ for some } j\in[n]\}$, the set of values that the $P$-partition $f$ attains on the non-$\widehat{0}$ elements of $\BB_n$. Set $\alpha = \#A$ and $\alpha(f) = \# A(f)$. For any $f,g\in\mathcal{F}_k(\BB_n)$, if $ g=w^j(f)$ for some $j\in\NN$, then $\alpha(f)=\alpha(g)$. So we may sometimes write just $\alpha$ when an orbit is fixed. For this section, we impose $A = A(f)$ when computing $\overline{w}:\mathcal{F}_k(C_n)\to\mathcal{F}_k(C_n)$ of an order-reversing map $f$.
\begin{ex}
Consider $f=(1,3,3,0,4,1,6)\in \mathcal{F}_9(\BB_6)$. We see $A(f) = \{0,1,3,4\}$ so 
$$\overline{w}(1,3,3,0,4,1,6) = (3,4,4,1,0,3,6).$$
\end{ex}
The last entry remains unchanged and the earlier entries are increasing cyclically within the set $A(f)=\{0,1,3,4\}$. In the special case where $\VV$ ($=\BB_2$) are set within any orbit $A$ will have at most two elements, hence $\overline{w}$ will just toggle between those two values at the left and the right. This means that $\overline{w}$ is the same as reflecting values across the center of $\VV$, which we already saw was the effect of $w^{k+2}$. Our next result generalizes this to the case $\BB_n$.

\begin{lem}\label{lem:Bconwhorm}
Let $f\in \mathcal{F}_k(\BB_n)$ and $\alpha = \alpha (f)$.  
If $\varsigma_1,\dots,\varsigma_{\alpha+1}$ are $\alpha+1$ consecutive whorms, then
$$\oc(\varsigma_1) + \cdots + \oc(\varsigma_{\alpha+1})  = \alpha(k+2).$$
\end{lem}



\begin{prop}\label{prop:worder}
Let $w$ be whirling $k$-bounded $P$-partitions on $\mathcal{F}_k(\BB_n)$. For any $f\in
\mathcal{F}_k(\BB_n)$ and $A=A(f)$, we have $w^{k+2}(f)=\overline{w}(f)$. 
\end{prop}

The proof of this theorem can be approached with whorms. 
Define $\oc(\varsigma) = 1+ \text{min}\{f(\hat 0): (\hat 0,f)\in \varsigma\}$.  
If there exists $(\hat 0, f) \in \varsigma_1$ with $f(\hat 0) = k$ such that $(\hat 0,
w(f))\in \varsigma_2$, then we say $\varsigma_2$ is \emph{in front of} $\varsigma_1$. In Figure $\ref{fig:Borbit}$, the pink snake is in front of the red snake.





\begin{wrapfigure}[7]{r}{85pt}
\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]

    \def\m{0}
    \tikzColoredSquare{0}{\m}{0}{white}
    \tikzColoredSquare{1}{\m}{1}{white}
    \tikzColoredSquare{2}{\m}{2}{white}
    \tikzColoredSquare{3}{\m}{3}{white}
    \tikzColoredSquare{4}{\m}{3}{white}
    
    \def\m{-1}
    \tikzColoredSquare{0}{\m}{1}{white}
    \tikzColoredSquare{1}{\m}{2}{white}
    \tikzColoredSquare{2}{\m}{3}{white}
    \tikzColoredSquare{3}{\m}{0}{white}
    \tikzColoredSquare{4}{\m}{3}{white}
    
    \def\m{-2}
    \tikzColoredSquare{0}{\m}{2}{white}
    \tikzColoredSquare{1}{\m}{3}{white}
    \tikzColoredSquare{2}{\m}{0}{white}
    \tikzColoredSquare{3}{\m}{1}{white}
    \tikzColoredSquare{4}{\m}{3}{white}
    
    \def\m{-3}
    \tikzColoredSquare{0}{\m}{3}{white}
    \tikzColoredSquare{1}{\m}{0}{white}
    \tikzColoredSquare{2}{\m}{1}{white}
    \tikzColoredSquare{3}{\m}{2}{white}
    \tikzColoredSquare{4}{\m}{3}{white}
\end{tikzpicture}
    \caption{}\label{fig:shortorbit}
\end{wrapfigure}

\begin{cor}\label{cor:Bwhormrepeat}
Let $f\in \mathcal{F}_k(\BB_n)$  and $\alpha = \alpha (f)$.  If $\varsigma_1,\dots,
\varsigma_{\alpha+2}$ are consecutive whorms, then $\oc(\varsigma_1) =
\oc(\varsigma_{\alpha+2})$.  
\end{cor}




If $f\in \mathcal{F}^k(\BB_n)$  satisfies $f(\widehat{0})\not\in A(f)$, then $f$ will
contain entries from $\alpha+1$ distinct whorms. From Proposition~\ref{prop:worder}, we will
have at most $\alpha(\alpha+1)$ whirls in an orbit board (each action of $\widehat{w}$
resulting in $\alpha+1$ whorms potentially distinct from those previous, as in
Figure~\ref{fig:Borbit}). On the other hands, consider the orbit
board of $\mathcal{F}_3(\BB_4)$ in Figure~\ref{fig:shortorbit} with $\alpha = 4$. 
Here $w(f) = w^{5}(f)$ so the orbit is only 4 rows long with 4 distinct whorms. In general,
we can extend this to a super orbit board with $\alpha(\alpha+1)$ whorms. 
\begin{thm}\label{thm:Bperiod}

Let $m=\min(k,n)$. The order of rowmotion on $\mathcal{J}(\BB_n\times [k])$ divides $m!(k+2)$.
\end{thm}




Only the analogue of the first homomesy in Theorem~\ref{thm:Vhom} holds.  

\begin{thm}\label{thm:homet} Let $\chi_{(i,j)}$ be the indicator function for $(i,j)\in
\BB_n\times [k]$. Then for the action of rowmotion on $\mathcal{J}(\BB_n\times [k])$, 
the statistic $\chi_{(i,a)}-\chi_{(j,a)}$ is $0$-mesic for all $i,j\in[n]$ and $a\in[k]$. 
\end{thm}

\begin{remark}
The average of the statistic $\left(\sum_{i=1}^n\chi_{(i,1)}\right)-\chi_{(\widehat{0},k)}$
(analogous to Theorem~\ref{thm:Vhom}(2))
turns out to be \textbf{dependent} on $\alpha(f)$ (for any $f\in \mathcal{O}$) and can be
computed as 
$$\frac{n(\alpha) (k+2) - (n+\alpha)(\alpha + 1)}{(\alpha)(k+2)}.$$


Consider the super-orbit with $n\alpha(k+2)$ entries among the non-minimal elements. We know $\chi_{(i,1)}(I)=0$ if and only if for corresponding $f$, $f(i) = 0$. But this is counted by the number of whorm beginnings, that is $n(\alpha+1)$. Furthermore, $\chi_{(\widehat{0},k)}(I)=1$ if and only if for corresponding $f$, $f(\widehat{0}) = k$, which is counted by the number of whorm endings, that is $\alpha(\alpha+1)$. Therefore the average is obtained.


\end{remark}

The ``flux-capacitor'' homomesy of Theorem~\ref{thm:flux} also generalizes to the claw-graph
setting, and has a similar proof.  
\begin{thm}\label{thm:Bflux}
Let $B_i = \chi_{(i-1,\widehat{0})}+ \sum_{j=1}^n \chi_{(i,j)}$.  Then for the action of rowmotion
on $\mathcal{J}(\BB_n\times [k])$,  $B_{i}-B_{j}$ is $\frac{(j-i)(n+1)}{k+2}$-mesic  for all
$i,j\in [n]$. 
\end{thm}

\acknowledgments{We are grateful to Banff International Research Station for hosting 
online and hybrid workshops on dynamical algebraic combinatorics in October 2020 and
November 2021. Discussions with the following colleagues were particularly helpful:
Sam Hopkins, 
Michael Joseph, 
James Propp, 
Bruce Sagan, 
and Jessica Striker. We also thank two anonymous reviewers of an earlier version of this work. }

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