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%% define your title in the usual way
\title[Symplectic Kostka--Foulkes polynomials I]{A positive combinatorial formula for symplectic Kostka--Foulkes polynomials I: Rows}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
        \author{Maciej Do\l{}\k{e}ga\thanks{\href{mailto:mdolega@impan.pl}{mdolega@impan.pl}. Supported by {\it Narodowe Centrum Nauki}, grant
   UMO-2017/26/D/ST1/00186. }\addressmark{1}, Thomas Gerber\thanks{\href{thomas.gerber@epfl.ch}{thomas.gerber@epfl.ch}. Supp. by the Ambizione project of the Swiss National Science Foundation.}\addressmark{2}, \and Jacinta Torres\thanks{\href{mailto:jtorres@impan.pl}{jtorres@impan.pl}. Supported by the { \it Deutsche Forscungsgemeinschaft} project TO 1135/1-1 and partially supported by  {\it Narodowe Centrum Nauki}, grant number 2017/26/A/ST1/00189 and the Max Planck Institute for Mathematics in the Sciences in Leipzig.  }\addressmark{1}} 
    
 %Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and \addressmark{2} \addressmark{3}}

%% then use \addressmark to match authors to institutions here

\address{\addressmark{1} Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland. \\ \addressmark{2} \'Ecole Polytechnique F\'ed\'erale de Lausanne, Lausanne, Switzerland }


%% 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{We prove a conjecture of Lecouvey, which proposes a closed, positive
combinatorial formula for symplectic Kostka--Foulkes polynomials, in
the case of rows of arbitrary weight. 
To show this, we transform the cyclage algorithm in terms of which the conjecture is described
into a different, more convenient combinatorial model, free of local constraints. 
In particular, we show that our model is governed by the situation in type A.
We expect our approach to lead to a proof of the conjecture in the general case. }

%% put your French abstract here, or comment this out if you don't have one
\resume{ Nous prouvons une conjecture de Lecouvey proposant une formule close positive pour les polyn\^omes de Kostka--Foulkes symplectiques, dans le cas des lignes de poids quelconque. Pour cela, nous transformons l'algorithme de cyclage gr\^ace auquel la conjecture est \'enonc\'ee en un mod\`ele combinatoire plus simple, sans contraintes locales. En particulier, nous d\'emontrons que notre mod\`ele est contr\^ol\'e par la situation en type $A$. Cette approche permet d'entrevoir une preuve de la conjecture dans le cas g\'en\'eral.   }

%% put your keywords here, or comment this out if you don't have them yet
 \keywords{combinatorial representation theory, Kostka--Foulkes polynomials, Lecouvey's conjecture, charge, Type C}

%% 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, ie. [xx, yy] in the text

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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}

The main motivation for this work is understanding an interplay
between combinatorics and representation theory which is highly
manifested in the structure of so-called \emph{Kostka--Foulkes polynomials}. 
Let $\mathfrak{g}$ be a complex, simple Lie algebra of rank $n$. Kostka--Foulkes polynomials $K_{\lambda, \mu}(q)$ are defined for two dominant integral weights as the transition 
% matrices 
coefficients
between two important bases of the ring of symmetric functions in the
variables $x = (x_1,..., x_n)$ over $\mathbb{Q}(q)$: Hall--Littlewood
polynomials $P_{\lambda}(x;q)$ and Weyl characters $\chi_\mu(x)$. They
are $q$-analogues of weight multiplicities \cite{Kato1982}, affine
Kazhdan--Lusztig polynomials \cite{Lusztig1983, Kato1982}, and appear
in various other situations in geometric and combinatorial
representation theory (see \cite{NelsenRam2003} and references
therein). We refer the reader to \cite{DolegaGerberTorres2019} for a precise definition
of Kostka--Foulkes polynomials and recommend \cite{NelsenRam2003} as a thorough reference.  \\

Due to their interpretation as Kazhdan--Lusztig polynomials, we know
that Kostka--Foulkes polynomials have nonnegative integer coefficients. This fact leads to one of the most important unsolved problems in combinatorial representation theory:


\begin{problem}
\label{problem}
Find a set $\cT(\lambda,\mu)$ and a combinatorial 
statistic
$\ch: \cT(\lambda,\mu) \to \Z_{\geq 0}$ such that the Kostka--Foulkes
polynomial $K_{\lambda,\mu}(q)$ is the generating function of
$\cT(\lambda,\mu)$ with respect to $\ch$. In other words find
$\cT(\lambda,\mu)$ and $\ch$ such that

\begin{align}
\label{formulaintro}
K_{\lambda,\mu}(q) = \sum_{T \in  \cT(\lambda,\mu)}q^{\ch(T)}.
\end{align}
\end{problem}

Since $K_{\lambda,\mu}(q)$ are  $q$-deformations of weight
multiplicities then $\#\cT(\lambda,\mu) = K_{\lambda,\mu}(1)$ is the
dimension of the $\mu$-weight space of the irreducible, finite
dimensional $\mathfrak{g}$-module of highest weight $\lambda$. In particular, in order to tackle
\Cref{problem} and find an appropriate set $\cT(\lambda,\mu)$, it
seems natural to seek for an object which parametrizes the aforementioned $\mu$-weight space of the irreducible, finite
dimensional $\mathfrak{g}$-module of highest weight $\lambda$. This
approach turned out to be very succesful in type $\AAA_{n-1}$, that is
when $\mathfrak{g} = \mathfrak{sl}(n,\mathbb{C})$. 
In this case dominant integral weights are identified with partitions of at most $n-1$ parts, and a natural candidate for $\cT(\lambda,\mu)$ is the set $\op{SSYT}(\lambda,\mu)$ of semistandard Young tableaux of shape $\lambda$ and weight $\mu$. 
In this context, Foulkes conjectured the existence of such a statistic
\cite{Foulkes1974}, which was explicitly found by Lascoux and
Sch\"utzenberger \cite{LascouxSchutzenberger1978}. They called their
statistic \emph{charge} (which explains our abbreviation $\ch$ used
also in the general context of arbitrary type)
and established the celebrated formula of \Cref{problem} in type $\AAA_{n-1}$. A thorough introduction to Kostka--Foulkes polynomials in type $\AAA_{n-1}$ and the charge statistic from a purely combinatorial point of view is carried out in \cite{Macdonald1995}. We  refer the reader to \cite{Butler1994} for a beautiful exposition and proof of \eqref{formulaintro}, which makes use of a recursive formula for computing Kostka--Foulkes polynomials due to Morris \cite{Morris1963}. 
 \\

In this work, we focus on \Cref{problem} for type $\CC_{n}$, that is, in case of the symplectic Lie algebra $\mathfrak{g} = \mathfrak{sp}(2n,\mathbb{C})$. 
To the best of our knowledge this is the only case of \Cref{problem}
having an explicit conjectural solution, which was formulated by
Lecouvey in \cite{Lecouvey2005}. In this case, the dominant integral
weights $\lambda,\mu$ can again be identified with partitions of at
most $n$ parts. The candidate for the set $\cT(\lambda,\mu)$ which features in Lecouvey's conjecture is
the set of \textit{symplectic tableaux}, which we will denote by $\op{SympTab}_{n}(\lambda,\mu)$, and which are also known as Kashiwara--Nakashima tableaux \cite{KashiwaraNakashima1994}.  These are defined to be
semistandard Young tableaux with some additional constraints (see
\cite{Lecouvey2005}) and entries in the ordered
alphabet \[\mathcal{C}_{n} = \{\overline{n} < \dots < \overline{1} <
  1 < \dots < n \}, \]
such that the shape of a tableau is given by $\lambda$ and its weight
by $\mu$. Here, the weight of a tableau with
entries in $\cC_n$ is defined slightly differently than the weight of a
tableau of type $\AAA_{n-1}$ and is given by the vector $(a_{\overline{n}}, ..., a_{\overline{1}})$, where $a_{\overline{i}}$
is the difference between the number of occurrences of
$\overline{i}$'s and $i$'s in $T$. Lecouvey defined a charge statistic
$\ch_{n}: \op{SympTab}_{n}(\lambda,\mu) \rightarrow \mathbb{Z}_{\geq
  n}$ by analogy with the situation in type $\AAA_{n-1}$, which makes use of the column insertion of a letter into a semistandard tableau. See the full version of this work \cite{DolegaGerberTorres2019} or  \cite{Lecouvey2005} for more details.
\\
Before we describe Lecouvey's conjectural solution to \Cref{problem}
involving cyclage it is worth mentioning that a solution to \Cref{problem} in type $C_n$ in the weight zero case has been given recently in \cite[Theorem 6.13]{LecouveyLenart2018},
using the combinatorial model for $\cT(\lambda,\mu)$ in terms of the so-called King tableaux introduced in \cite{King1976}.
However, this relies on an interpretation of the Kostka--Foulkes
polynomials in terms of generalized exponents which only holds in this
special case of weight zero,
so that there is little hope to tackle the general weight case with this approach.


\subsection{Main result and methodology}In order to define the statistic $\op{ch}_{n}: \op{SympTab}_{n}(\lambda, \mu) \rightarrow \mathbb{N}$ and formulate his conjecture, Lecouvey used a symplectic version of column insertion, which he introduced in \cite{Lecouvey2002}, to define a symplectic cyclage operation $\op{Cyc}_{\CC}$ which transforms a symplectic tableau $T \in \op{SympTab}_{n}$ into a symplectic tableau $\Cyc_{\CC}(T) \in \op{SympTab}_{m}$ for 
$m\geq n$.
The statistic $\op{ch}_{n}$ is defined as follows. Let $T \in \op{SympTab}_{n}$ be a symplectic tableau. In \cite{Lecouvey2005}, Lecouvey showed that there exists a non-negative integer $m$ such that $\Cyc_{\CC}^{m}(T)$ is a column $C(T)$ of weight zero. We denote by $m(T)$ the smallest non-negative integer with this property. For a symplectic column $C$ of weight zero we set $E_{C}= \{ i \geq 1| i \in C, i+1 \notin C\}.$ The charge of $C$ is defined by 


\[ \op{ch}_{n}(C) = 2 \underset{i \in E_{C}}{\sum} (n-i),\]
and the charge of an arbitrary symplectic tableau $T$ is defined by 

\[\op{ch}_{n}(T) = m(T) + \op{ch}_{n}(C(T)).  \]Lecouvey
\cite{Lecouvey2005} conjectured the following solution of \Cref{problem} in type $\CC_{n}$:
\begin{conj}
\label{conji}
Let $\mu,\lambda$ be partitions with at most $n$ parts. Then 
\begin{align}
\label{conj}
 K^{\CC_n}_{\lambda,\mu}(q) = \underset{T \in \op{SympTab}_{n}(\lambda,\mu)}\sum q^{\op{ch}_{n}(T)}.    
\end{align}
\end{conj}

Our main result reads as follows.

\begin{theorem}
\label{mainresultintro}
Let $\lambda = (p)$ and $\mu = (\mu_{\ov n},\dots,\mu_{\ov 1})$ be an
arbitrary partition. Then \Cref{conji} holds true:
\[K^{\CC_{n}}_{\lambda,\mu}(q) = \sum_{T \in
    \op{SympTab}_{n}(\lambda,\mu)} q^{\op{ch}_{n}(T)}. \]
\end{theorem}

A pivotal point in our methodology, and one which we expect will have impact on the study of the general case of \Cref{conji}, 
is a reformulation of Lecouvey's construction in the setting of
\Cref{mainresultintro} by
providing a new algorithm to compute $\Cyc^k_{\CC}(T)$ for
arbitrary integer $k$. The
big advantage of our approach is that in \Cref{alg:content}, which
completes this task, we are able to eliminate
local constraints which appear in the original construction in two
different contexts:
\begin{itemize}
  \item[$\bullet$] we need to compute $\Cyc^{k-1}_{\CC}(T)$ in order to compute
    $\Cyc^{k}_{\CC}(T)$;
    \item[$\bullet$] for each column of $\Cyc^{k-1}_{\CC}(T)$, we need to insert boxes recursively into
      consecutive subcolumns of size $2$.
    \end{itemize}
In order to eliminate the second constraint we give a formula for
inserting an entry into a whole column at once, which is given by \cite[Proposition~3.3]{DolegaGerberTorres2019}. 
Although more technical in appearance, our new definition allows us to
have a full grasp of the symplectic cyclage procedure. We show in
\Cref{prop:crucial} that for a partition $\lambda = (p)$ which consists of one row
and for an arbitrary partition $\mu$ the symplectic tableau $\Cyc^{k}_{\CC}(T)$, where $T \in
\op{SympTab}_{n}(\lambda,\mu)$, is given by \Cref{alg:content}. The
main philosophy of \Cref{alg:content} is that in order to compute
$\Cyc^{k}_{\CC}(T)$, it is enough to only apply $\Cyc_{\AAA}$ to certain
standard Young tableaux and then apply a very simple function
which changes the entries of the outcome. \\

As an application, we are able to
compute $\op{ch}_{n}(T)$ directly from $T$ and, using a
simple recurrence for Hall--Littlewood polynomials of type $\CC$ proved
by Lecouvey in \cite[Theorem 3.2.1.]{Lecouvey2005}, we deduce
\Cref{mainresultintro}. We believe that our strategy might lead
us to the solution of \Cref{conji} in the full generality. Indeed, the restriction $\lambda = (p)$ is due to the
fact that symplectic tableaux of row shape coincide with semistandard
tableaux with entries in the alphabet $\mathcal{C}_{n}$ (see
\Cref{prop/def}). In particular, there exists a unique standard tableau of
shape $(p)$, and \Cref{alg:content} consists in applying $\Cyc_{\AAA}$ multiple
times to this unique tableau. It seems likely that in the more general
case there exists a ``right''
labelling of the boxes of any symplectic tableau $T$ of arbitrary shape,
such that a very similar procedure could be followed to compute $\Cyc_{\CC}^k(T)$
and therefore $\op{ch}_{n}$. So far, this question remains open and we will be investigating this question in the future. The full version of this work, \cite{DolegaGerberTorres2019}, which contains the full proofs of all the results presented in this extended abstract,  will be submitted elsewhere.

\subsection{Notation}
\emph{A composition} $\alpha \vDash n$ of size $n \in \Z_{\geq 0}$ is a
sequence of non-negative integers $\alpha := (\alpha_1,\alpha_2,\dots)
\in \Z_{\geq 0}^{\Z_{>0}}$ such that $\sum_{i}\alpha_i = n$ and such
that $\alpha_i=0$ implies that
$\alpha_{i+1}=0$ for any $i \in \Z_{>0}$  In particular, there are
only finitely many non-zero $\alpha_i$ and we denote their number by
$\ell(\alpha)$ calling it the \emph{length} of  composition $\alpha$. We will also use the notation $|\alpha| = n$. A \textit{partition} is a composition with non-decreasing entries. We denote the set of compositions of size $n$ by $\Comp_n$, the set of partitions of size $n$ by $\Part_{n}$ and we set $\Comp := \bigcup_{n}\Comp_n$, respectively $\Part := \bigcup_{n}\Part_n$. To any $\alpha \in \Comp_n$ we associate its diagram defined by $ \cD_\alpha = \{(i, j):1 \leq i \leq \alpha_{-j}, j \leq -1 \} \subset \mathbb{Z}_{>0} \times \mathbb{Z}_{<0}.$ The elements of $\cD_\alpha$, referred to as boxes, are linearly ordered by the so-called \emph{reading order}, which is a variant of lexicographic order: $(i_1,j_1) \leq (i_2,j_2)
\iff j_1 > j_2$ or $j_1 = j_2, i_1 < i_2$. For an ascending chain $(\calA,\prec)$, that is a linearly ordered alphabet with minimal element $a$ and for $b \in \calA$ we denote the $b$-th box of $\cD_\alpha$ in reading order by $\square_b$, or by $b$ whenever it does not lead to a confusion. For any composition $\alpha \vDash n$ we define \emph{a tableau} $T$ of
shape $\alpha$ and entries in $\calA$ to be a filling of the boxes of the diagram of $\alpha$ by elements from alphabet $\calA$, that is a map $T: \cD_\alpha \rightarrow \mathcal{A}.$
The \emph{content} of a tableau $T$ of shape $\alpha$ is the multiset of its entries. When $\calA$ is a countable ascending chain (with the minimal element $a$), we say that a tableau has \emph{weight} $\beta
= (\beta_1,\beta_2,\dots)$ when its content is given by the multiset $\{a^{\beta_1},(a+_\succ)^{\beta_2},\dots, (a+_{\succ^k})^{\beta_{k+1}},\dots\},$ where $a+_{\succ}$ denotes the successor of $a$, and $a+_{\succ^{k+1}}:= a+_{\succ^k}+_{\succ}$. We denote the set of semistandard Young tableaux of shape $\alpha$ and weight $\beta$ with entries in $\mathcal{A}$ by $\op{SSYT}_{\mathcal{A}}(\alpha, \beta)$. 


%\subsection{Symplectic cyclage and charge}

Let $n$ be a positive integer and $\lambda,\mu$ partitions with at most $n$
parts.  The following proposition justifies why we do not need the
defining properties of symplectic tableaux in this work. It is a direct consequence of Definition in  \cite{Lecouvey2005}. 
\begin{proposition}
\label{prop/def}
Let $\lambda = (p)$ and $\mu$ be a partition. Then 
\[\op{SympTab}_{n}(\lambda, \mu) = \bigcup_{k_1,\dots,k_n \in \Z_{\geq
    0}}\op{SSYT}_{\mathcal{C}_n}(\lambda, (k_n+\mu_{\ov{n}},k_{n-1}+\mu_{\ov{n-1}},\dots,
  k_{1}+\mu_{\ov{1}},k_1,\dots,k_n)). \]
% \noindent
% That is, in the case of $\lambda = (p)$, admissibility conditions are empty.
\end{proposition}

 We are interested in the set of symplectic tableaux since these
objects give a natural basis of the $\mu$ weight
space of an irreducible $\mathfrak{g}$--module of highest weight
$\lambda$ in type $C$. In particular, $K^{C_n}_{\lambda,\mu}(1) = |\op{SympTab}_{n}(\lambda,\mu)|.$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The shift and content algorithms.}

In this section we will construct the new algorithm computing
$\Cyc^{k}_{C}(T)$ for arbitrary $k>0$ and for $T \in \SympTab((p))$, that is $T$ is 
a symplectic tableau of row shape. 
Our algorithm does not rely on the particular form of
$\Cyc^{k-1}_{C}(T)$, which allows us to overcome the problem of
controlling many
local dependencies present in Lecouvey's original algorithm.
This construction is the main ingredient in the proof of \Cref{mainresultintro}. Reordering the parts of a composition $\al\vDash n$ gives a partition $\la\vdash n$.
Note that $\la$ can be also seen as the result of lifting all the
boxes in each column of $\al$ so that after the lift, the boxes in the given column are lying in consecutive rows starting from the first row.
For this reason, we denote by $\grav$ the map $\Comp_n\to\Part_n, \al\mapsto\la$
and call it the gravity map. This map is naturally also defined on tableaux. 

%\begin{exa}
%\Yboxdim{1cm}
%We have
%$
%\grav \left(
%\begin{tikzpicture}[font=\tiny, scale=0.3,baseline=0cm]
%\tgyoung(0cm,1cm,<1>,<2><3>,<4><4><5><6>,<5>)
%\end{tikzpicture}\right)
%=
%\begin{tikzpicture}[font=\tiny, scale=0.3,baseline=0cm]
%\tgyoung(0cm,1cm,<1><3><5><6>,<2><4>,<4>,<5>)
%\end{tikzpicture}
%$.
%\end{exa}

\subsection{Shifting}
\label{section:shifting}

 $n\in\Z_{\geq0}$ and define $\shift: \Comp_n \to \Comp_n$
as follows 
\[ \shift(\alpha) = \begin{cases} \alpha &\text{ if $\alpha = (1^l,0,\dots)$ for
      some $l \in \Z_{\geq 0}$};\\
\alpha - e_i+e_{i+1} &\text{ otherwise};\end{cases}\]
where $e_i = (\underbrace{0,\dots,0}_{\text{$i-1$ times}},1,0\dots)$
and $i = \min\{j \mid \alpha_j = \max_k \alpha_k\}$. We define the operator
\begin{align*}
\op{simp}:\Comp\times\Part
\to \Comp\times\Part
\end{align*}
by the following recursive algorithm: 

\begin{algorithm}[H]
\caption{Defining $\op{simp} (\alpha,\mu)$.}
\label{alg:red}
\begin{algorithmic} 
\Require A partition $\mu$ and a composition $\alpha$.
\Ensure A pair $(\beta,\nu) \in \Comp\times\Part$.\\
$\beta = \alpha$\\
$\nu = \mu$
\While{$\max \beta_{k} = \nu_{1}$}
  \State $\nu = \nu \setminus \nu_{1}$
  \State $\beta = \beta \setminus
   \max \beta_{k}$
\EndWhile
\end{algorithmic}
\end{algorithm}

We extend the domain of the operator $\shift: \Comp\times\Part
\to \Comp\times\Part$ by: 

\[ \shift(\alpha,\mu) = \begin{cases} (\shift(\alpha),\mu) &\text{ if
      $(\alpha,\mu) \neq \op{simp} (\alpha,\mu)$};\\
 \left(\shift\big(\op{simp} (\alpha,\mu)_1\big), \op{simp}(\alpha,\mu)_2\right) &\text{ otherwise};\end{cases}\]
where $\op{simp} (\alpha,\mu)_i$ denotes the $i$-th coordinate of $\op{simp} (\alpha,\mu)$. For the geometric intuition behind these definitions, see \cite[Section~2.4]{DolegaGerberTorres2019}.


\begin{lemma}\cite[Lemma~2.6]{DolegaGerberTorres2019} 
\label{lem:ShiftTerminates}
For any pair $(\alpha,\mu) \in \Comp\times\Part$ there exists an
integer $m$ and a partition $\nu$ such that $\shift^{m}(\alpha, \mu) = ((1^l), \nu)$ and is a fixed
point of $\shift$ (for some $l
\geq 0$), that is $\nu_1 \neq 1$.
\end{lemma}

We define 

\begin{equation}
  \label{eq:minimal}
  m_{\mu}(\alpha) = \min \{m | \shift^{m+1}(\alpha,\mu) =
  \shift^{m}(\alpha,\mu)\}.
  \end{equation}

\begin{corollary} \cite[Corollary~2.7]{DolegaGerberTorres2019}.\label{numsteps}
In the special case $\alpha = (p), |\mu|\leq p$ we have 
\[ m_{\mu}(\alpha) = \sum_i(i-1)\mu_i + \frac{(p-|\mu|)(p-|\mu| + 2\ell(\mu)-1)}{2}.\]
\end{corollary}



\subsection{The content function.}

Given a composition $\alpha$ and  two boxes $b$ and $b'$ in $\alpha$ such that $b < b'$ in the reading order, their \textit{distance} in $\alpha$
is defined by
\begin{align*}
\delta_{\alpha}(b, b') = \row_{\alpha}(b') - \row_{\alpha}(b) - \chi\big(\col_{\alpha}(b) \geq \col_{\alpha}(b')\big),
\end{align*}
where, for a condition $\sC$, we define $\chi(\sC) = 1$ if  $\sC$ is satisfied, and $\chi(\sC) = 0$ otherwise, and $\row_{k}(s)$ and $\col_{k}(s)$ denote the row and column index of $s$ counted from top to bottom and from left to right, respectively. In \Cref{alg:content} we define a tableau which we show to be equal to $\Cyc^{k}(T)$. Note that \Cref{alg:content} decomposes the set of boxes
into two disjoint sets. The first set contains the boxes which had no
associated partners; we call such a box $b$ a \emph{single}. All the
other boxes are matched into pairs by associating their partners; for
such a box $b$ we denote by $\partner(b)$ its partner (note that
$\partner(\partner(b)) = b$).

\begin{exa}[Weight zero]
\label{example:weightzero}
Let $T$ be a tableau of shape $(2q)$ and weight zero (note that all tableaux of weight zero must have an even number of boxes). We may label its boxes by elements in the interval $[\ov{q},q]  \subset \mathcal{C}$. We have $\alpha = \shift^{k}((2q),0)_{1}  = \shift^{k}((2q))$, and the content of a given box in $T_{k}$ is given by

\[T_{k}(S) = \begin{cases} T(S) +
  \delta_{\alpha}(\overline S, S) &\text{ if } S >0,\\
\overline{T_{k}(\overline S)} &\text{ if } S <0.
\end{cases}\]

Boxes $\overline{S}$ and $S$ are always \textit{partners}, and they will hence have opposite contents in $T_{k}$ for each $k\leq m(T)$.
\end{exa}

\begin{algorithm}[H]
\caption{Defining the tableau $T_{k}$.}
\label{alg:content}
\begin{algorithmic} 
\Require Nonnegative integers $k,k_1,\dots,k_n$ and a partition $\mu=(\mu_{\ov{n}},\dots,\mu_{\ov{1}})$.
\Ensure The tableau $T_{k}: [1,|\al|]\ra \cC$ of shape $\al$.
\State $p = \sum_{i = 1}^n(2k_i+\mu_{\ov{i}})$
\State $\al = \shift^{k}((p),\mu)_{1}$
\State $\nred = \ell(\mu)-\ell(\shift^{k}((p),\mu)_{2})$
\Comment{$\nred$ counts the number of reductions performed so far}
\State $R = n-\nred +1$
\State $I_\al =
         \{\overline{n+\nred}^{k_n},\dots,\overline{r+\nred}^{k_r},
           \overline{r-1+\nred}^{k_{r-1}+\mu_{\ov{r-1}}}, \dots,
           \overline{1+\nred}^{k_1+\mu_{\ov{1}}},
         (1+\nred)^{k_1},\dots,(n+\nred)^{k_n}\}$
\State $f_{\alpha}:[1,|\al|]\to I_\al$ s.t. $f_\al$ is the unique
non-decreasing bijection
\Comment{$f_\al$ is the natural labeling of elements in the multiset $I_\al$}
\State $D = \min\{ S\in[1,|\al|]\mid f_\al(S) \text{ is unbarred}\}$
\Comment{$D = \sum_i^n k_i + \sum_i^{r-1}\mu_{\ov{i}}+1$}
\State $D' = \max\{ S\in[1,|\al|]\mid f_\al(S) \text{ is barred}\}$
\Comment{$D' = \sum_i^n k_i + \sum_i^{r-1}\mu_{\ov{i}}$}
\State $M = 1$
\While{$D\leq |\al|$}
\State $\partners = \false$
    \State $X = f_\al(D)+\de_\al(D',D)$
    \While{$\partners == \false$} 
        \If{$X<M+\nred$ or $M\geq R$} 
            \State $\partners = \true$
            \State $T_{k}(D') = \ov{X}$, $T_{k}(D)=X$
            (the boxes $D'$ and $D$ are said to be \textit{partners})
            \State $D=D+1$, $D'=D'-1$
        \Else
            \State $T_{k}(S)=\ov{M+\nred}$ for all $S\in[D'-\mu_{\ov{M}}+1,D']$
            \State $D'=D'-\mu_{\ov{M}}$
            \State $M=M+1$
        \EndIf
    \EndWhile
\EndWhile
\end{algorithmic}
\end{algorithm}


\begin{exa}
\label{example:contentnored} 

\[\hbox{Let }T = \Skew(0: \hbox{\scriptsize{$\ov{3}$}}, \hbox{\scriptsize{$\ov{3}$}},\hbox{\scriptsize{$\ov{3}$}},\hbox{\scriptsize{$\ov{3}$}}, \hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{2}$}},\hbox{\scriptsize{$\ov{2}$}},\hbox{\scriptsize{$\ov{2}$}},\hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{1}$}}, \hbox{\scriptsize{$\ov{1}$}},\hbox{\scriptsize{$\ov{1}$}},\hbox{\scriptsize{$\ov{1}$}},\hbox{\scriptsize{$\ov{1}$}}, \hbox{\scriptsize{1}}, \hbox{\scriptsize{1}}, \hbox{\scriptsize{1}}, \hbox{\scriptsize{1}},\hbox{\scriptsize{2}}, \hbox{\scriptsize{2}}, \hbox{\scriptsize{2}} ,\hbox{\scriptsize{3}}). \]

Then $n=3$, $(k_3,k_2,k_1) = (1,3,4)$ and $\mu =
(\mu_{\ov{3}},\mu_{\ov{2}},\mu_{\ov{1}}) = (3,2,1)$. We would like to
compute $T_{78}$.  Since $p= 22$, we know that $\alpha =
\shift^{78}((22),\mu)_{1}$ which is equal to $(2,2,3,4,3,3,2)$. Moreover
$\shift^{78}((22),\mu)_{2} = (2,1)$, thus $\nred = 1$, $R = 3$ and $I_\al = \{\ov{4}, \ov{3}^{5}, \ov{2}^{5}, 2^4,3^3,4\}$.


Let us first assign labels to all the boxes in $\alpha$ according to the reading order: 

\[\Skew(0:\hbox{\scriptsize{1}},
  \hbox{\scriptsize{2}}|0:\hbox{\scriptsize{3}},\hbox{\scriptsize{4}}
  |0: \hbox{\scriptsize{5}},\hbox{\scriptsize{6}},
  \hbox{\scriptsize{7}}|0: \hbox{\scriptsize{8}},
  \hbox{\scriptsize{9}}, \hbox{\scriptsize{10}},
  \hbox{\scriptsize{11}}|0:
  \hbox{\scriptsize{12}},\hbox{\scriptsize{13}},
  \hbox{\scriptsize{14}}|0:\hbox{\scriptsize{15}},\hbox{\scriptsize{16}},\hbox{\scriptsize{17}}|0:
  \hbox{\scriptsize{18}}, \hbox{\scriptsize{19}}).\]

Note that at the beginning of our algorithm $D=12, D'=11$ and
$M=1$.
We perform the algorithm described above to find a partner box for
${12}$ in $\alpha$ and to calculate their contents. Since
\[\delta_{\alpha}({11}, {12}) = 0 ,\]
\noindent 
we have that $X = f_\al(12) +
\delta_{\alpha}({11}, {12})  = 2 = M+\nred$. Therefore
${12}$ and ${11}$ are not partners, and $
T_{78}({11}) = \ov{2}, D'=10,M=2$. Thus $11$ is a single and we are still looking
  for a partner for ${12}$. $X = f_\al(12) +
\delta_{\alpha}({10}, {12})  = 2 < 3 = M+\nred$ now, which means that
$\partner(12) = 10$ so
$\ov{T_k({12})} = T_k({10}) = 2$ and
$D=13,D'=9$. Similarly as before $\partner(13) = 9$ so $\ov{T_k({13})} = T_k({9}) = 2$ and
$D=14,D'=8$. At this step our tableau has the following form:
\[\Skew(0:,|0:,|0:,, |0:,\hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{2}$}}|0: \hbox{\scriptsize{$2$}}, \hbox{\scriptsize{$2$}},|0:,, |0: ,).\]
Note that now
\[\delta_{\alpha}({8}, {14}) = 1,\]
so ${8}$ and ${14}$ are not partners since $X = f_\al(14) +
\delta_{\alpha}({8}, {14})  = 2 + 1 =
M+\nred$. Therefore $8$ and $9$ are singles and $T_{78}({7}) = T_{78}({8}) =
\ov{3}, D'=6 ,M=3$. But now $M\geq 3 = R$ so our algorithm will assign
partners at every step: $({14},{6}),
({15},{5}), ({16},{4}),
({17},{3}), ({18},{2}),
({19},{1})$. Moreover, the distances
between partners are as follows:
\begin{align*}
\delta_{\alpha}({6},{14}) =
  \delta_{\alpha}({5},{15}) &= 2, \\
\delta_{\alpha}({4},{16}) &=3, \\
\delta_{\alpha}({3},{17}) &=4,\\ 
\delta_{\alpha}({2},{18}) &= 5,\\
\delta_{\alpha}({1},{19}) &=6. 
\end{align*}
Since
\begin{align*}
f_{\al}(6) = f_{\al}(5) &= 2, \\
f_{\al}(4) = f_{\al}(3)
=f_{\al}(2) &=3, \\
f_{\al}(1) &=4,
\end{align*}
we obtain the following tableau $T_{78}$:

\[\Skew(0: \hbox{\scriptsize{$\ov{10}$}}, \hbox{\scriptsize{$\ov{8}$}}|0: \hbox{\scriptsize{$\ov{7}$}}, \hbox{\scriptsize{$\ov{6}$}}|0: \hbox{\scriptsize{$\ov{4}$}}, \hbox{\scriptsize{$\ov{4}$}}, \hbox{\scriptsize{$\ov{3}$}} |0: \hbox{\scriptsize{$\ov{3}$}},\hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{2}$}}, \hbox{\scriptsize{$\ov{2}$}}|0: \hbox{\scriptsize{$2$}}, \hbox{\scriptsize{$2$}}, \hbox{\scriptsize{$4$}}|0: \hbox{\scriptsize{$4$}}, \hbox{\scriptsize{$6$}}, \hbox{\scriptsize{$7$}} |0: \hbox{\scriptsize{$8$}} , \hbox{\scriptsize{$10$}}).\]

\end{exa}

In \cite{DolegaGerberTorres2019}, we show the following proposition, which is the key ingredient in the proof of our main result. 

\begin{proposition}
\label{prop:crucial}
Let $\mu = (\mu_{\ov{n}},\mu_{\ov{n-1}},\dots,\mu_{\ov{1}})$ be a partition, $\ell(\mu) = \#\{i\colon \mu_i > 0\}$, and let $T \in \op{SympTab}_{n}((p),\mu)$. Then 

\begin{align}
\label{crucialfin}
 \Cyc^{k}_{C}(T) = \red(\grav (T_{k})),
 \end{align}
 
 \noindent
 where the reduction operation is as defined in \cite[Section~4.3]{Lecouvey2005}, see also \cite[Section~3.4]{DolegaGerberTorres2019}. 
\end{proposition}

\begin{proof}[Idea of proof]
The idea of the proof is to carry out a careful, recursive study of the cyclage, and in particular insertion algorithms of Lecouvey. In particular, we need a slightly different definition of Lecouvey's insertion algorithm, which can be found in \cite[Proposition~3.3]{DolegaGerberTorres2019}. Although the idea of the proof is simple, it is rather involved due to the technical nature of the symplectic insertion algorithm. The reader is therefore referred to the full version of this work, see \cite[Section~4]{DolegaGerberTorres2019}. 
\end{proof}


\section{A proof of Lecouvey's conjecture for one row and general weight}

In this section we are going to apply \eqref{crucialfin} to prove
\Cref{mainresultintro}. We need the following proposition
due to Lecouvey, which is an easy consequence of the following Morris recurrence formula described in \cite{Lecouvey2005}: 

\begin{proposition}{\cite [Proposition 3.2.3.]{Lecouvey2005}}
 \label{onerowlecouvey}
Let $\mu = (\mu_{\ov{n}},\mu_{\ov{n-1}},\dots,\mu_{\ov{1}})$ be a
partition and $p \geq |\mu|$ be a positive integer. Then 
\[ K^{\CC_n}_{(p),\mu}(q) = q^{f_n(\mu)}\cdot\sum_{T \in \SympTab_n((p),\mu)} q^{\theta_n(T)}\]
where $f_{n}(\mu) = \sum_{i=1}^{n}(n-i)\mu_{\ov i}$ and 
\[\theta_{n}(T) = \sum_{i=1}^{n}(2(n-i)+1)k_i,\]
where $T \in \SSYT_{\mathcal{C}_n}((p), (k_n+\mu_{\ov{n}},k_{n-1}+\mu_{\ov{n-1}},\dots,
  k_{1}+\mu_{\ov{1}},k_1,\dots,k_n))$.
\end{proposition}
   
  \begin{proof}[Proof of \Cref{mainresultintro}]
Let $T$ be as in the statement of \Cref{onerowlecouvey}. \Cref{prop:crucial} (see \cite[Section~3.4]{DolegaGerberTorres2019}) implies that
  $m(T) = \min\{k: T_k = T_{k+1}\}$, which is simply equal to
  $m_{\mu}((p))$ defined by \eqref{eq:minimal}. Let us compute $E_{C(T)}$. Notice that $C(T)$ is a
column of weight $0$ and length $\sum_ik_i$. Therefore, for any
$\square,\square+1 \in I_{>0}$ we have
\begin{align*} C(T)(\square+1)-C(T)(\square) &=
                                               \delta_{\shape(C(T))}\big(\partner(\square+1),\square+1\big)-\\
  &-\delta_{\shape(C(T))}\big(\partner(\square),\square\big)=2.
  \end{align*}
Therefore $E_{C(T)}$ consists of all positive entries of $C(T)$ and
due to the construction given by \Cref{alg:content} we know that
$\nred = \ell(\mu)$, thus
\[ E_{C(T)} = \{i+\ell(\mu)+2j\colon 1\leq i \leq n, \sum_{l \leq i-1}k_l \leq j < \sum_{l \leq i}k_l\}.\]
 Finally, using  \Cref{numsteps} we may calculate (see \cite{DolegaGerberTorres2019}):
 \begin{align*} 
\ch_n(T) =  f_n(\mu)+\theta_n(T)
 \end{align*}
and comparing this with \Cref{onerowlecouvey} finishes the proof.
\end{proof}

\section*{Acknowledgments}
We thank C\'edric Lecouvey for many interesting
conversations. The SageMath computer algebra system
\cite{sagemath} has been used for experimentation leading up to many of the results
presented in the paper.


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