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\title{Colored Permutation Statistics by Conjugacy Class}

\author[Campion Loth, Levet, Liu, Sundaram, and Yin]{
Jesse Campion Loth\thanks{\href{mailto:jesse_campion_loth@sfu.ca}{jesse\_campion\_loth@sfu.ca}}\addressmark{1}, 
Michael Levet\thanks{\href{mailto:levetm@cofc.edu}{levetm@cofc.edu}. Partially supported by J. A. Grochow's NSF award CISE-2047756.}\addressmark{2},
Kevin Liu\thanks{\href{mailto:kliu15@uw.edu}{kliu15@uw.edu}}\addressmark{3},
Sheila Sundaram\thanks{\href{mailto:shsund@comcast.net}{shsund@umn.edu}}\addressmark{4},
\and Mei Yin\thanks{\href{mailto:mei.yin@du.edu}{mei.yin@du.edu}. Partially supported by the University of Denver's Professional Research Opportunities for Faculty Fund 80369-145601.}\addressmark{5}
}

\address{
\addressmark{1}Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada \\ 
\addressmark{2}Department of Computer Science, College of Charleston, Charleston, SC, USA \\
\addressmark{3}Department of Mathematics, University of Washington, Seattle, WA, USA \\
\addressmark{4}University of Minnesota, Minneapolis, MN, USA \\
\addressmark{5}University of Denver, Denver, CO, USA
}

\received{October 28, 2023}
\revised{March 20, 2024}

\abstract{We consider the moments of statistics on conjugacy classes of colored permutation groups $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We first show that any fixed moment of a statistic coincides on all conjugacy classes when all cycle lengths are sufficiently long. For permutation statistics that can be realized via a process called symmetric extension, we show that for fixed $r$, this moment on these conjugacy classes is a polynomial in $n$. Hamaker and Rhoades (arXiv, 2022) established analogous results for the symmetric group as part of their far-reaching representation-theoretic framework. Independently, Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin (arXiv, 2023) arrived at independence and polynomiality results for the symmetric group using instead an elementary combinatorial framework. Our techniques in this paper build on this latter elementary approach. Finally, we extend the work of Fulman (\textit{J. Comb. Theory Ser. A.}, 1998), to establish a central limit theorem for descents in conjugacy classes of the hyperoctahedral group with sufficiently long cycles.}

\keywords{colored permutation, Coxeter group, hyperoctahedral group, moment, permutation constraint, permutation statistic}

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\begin{document}

\maketitle

\section{Introduction}
For a finite group $G$, a \textit{statistic} is a map $X : G \to \mathbb{R}$. The \textit{distribution} of $X$ is the function $(x_{k})$, where $x_{k}$ is the number of elements $g \in G$ such that $X(g) = k$ (i.e., $x_{k} := |X^{-1}(k)|$). When $G$   is the symmetric group $G = \mathfrak{S}_{n}$, we refer to the statistics as \textit{permutation statistics}. The study of permutation statistics is a classical topic in algebraic combinatorics; Stanley's texts \cite{StanEC1, StanEC2} serve as a key reference in this area.

In this paper, we build on the elementary methods  in \cite{GRWCPermutationStatistics} to investigate the distribution of \emph{colored} permutation statistics by conjugacy class.
In contrast to the vast literature on permutation statistics in $\mathfrak{S}_{n}$, there has been considerably less work on statistics for arbitrary Coxeter groups or the colored permutation groups $\mathfrak{S}_{n,r}$, i.e., the wreath product $\mathbb{Z}_r\wr\mathfrak{S}_n$. We are in particular not aware of work considering colored permutation statistics on individual conjugacy classes.

When $r=2$, the colored permutation group $\mathfrak{S}_{n,2}$ coincides with the hyperoctahedral group $B_n$, which is the type $B$ Coxeter group. A study of statistics over the entire Coxeter group for  types B and D was initiated by Reiner, see e.g. \cite{ReinerEuropJC1993-2}, 
and carried further by Adin and Roichman, see e.g. \cite{AdinBrentiRoichman}, 
and Brenti and Carnevale \cite{BrentiCarnevaleOddLength}.
There is also work on colored permutation statistics and their distribution, again over the whole group, by Steingr\'{i}msson \cite{steingrim94}, Fire \cite{ fire2005statistics}, and Moustakas \cite{MOUSTAKAS2021102177}. 

Recently, Hamaker and Rhoades \cite{hamaker2022characters} established a representation-theoretic framework for permutation statistics on $\mathfrak{S}_{n}$ by conjugacy class $C_{\lambda}$. They introduced so-called \textit{local} permutation statistics;  using representation-theoretic methods, they established that the moments of these statistics depend only on $n$ and the number of short cycles in $\lambda$. In particular, these moments are independent of the conjugacy class when the cycles in $\lambda$ are all sufficiently large. 

Independently, and subsequent to the paper \cite{hamaker2022characters}, 
Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin \cite{GRWCPermutationStatistics} established similar independence and polynomiality results for conjugacy classes in $\mathfrak{S}_n$, 
using only elementary combinatorial techniques. The present paper builds on the framework in \cite{GRWCPermutationStatistics}.  The full version of this paper appears in \cite{GRWCColoredPermutationStatistics}.

\noindent \\ \textbf{Main Results.} Fix $r \geq 1$, and let $\rpart$ be an $r$-partition of $n$. For a statistic $X$ on $\mathfrak{S}_{n,r}$, denote by  $\E_{\rpart}[X]$  the expected value of $X$ taken over the conjugacy class of $\mathfrak{S}_{n,r}$ indexed by $\rpart$. Our  main results are as follows:
\begin{itemize}
 \item Theorem~\ref{thm:IndependenceHigherMoments} in Section~\ref{sec:Indep-higher-mom} shows that for any statistic $X$, its $k$th moment coincides on all conjugacy classes $C_{\rpart}$ of $\mathfrak{S}_{n,r}$ that do not have "short" cycles. For each statistic $X$, making this notion of "short" precise is done through \emph{colored permutation constraints} as given in Definition~\ref{def:ColoredPermutationConstraints}. 
 
 \item Theorem~\ref{thm:HigherMomentPolynomial} in Section~\ref{sec:symmetric-colored-perms}
 concerns sequences of statistics $(X_n)_{n\geq 1}$  on $(\mathfrak{S}_{n,r})_{n\geq 1}$ that can be constructed using \emph{symmetric extensions}, as described in Definition~\ref{def:SymmetricExtensions}. This theorem shows that a single polynomial in $n$ gives $\E_{\rpart_n}[X_n^k]$ on conjugacy classes $C_{\rpart_n}$ of $\mathfrak{S}_{n,r}$ without "short" cycles. Note that this result applies to many statistics, including the inversion statistic on $B_n$ defined in \eqref{eqn:inv-negsum}.


 \item Finally, Theorem~\ref{thm:CLTMainS} in Section \ref{section:CLT} %\meisay{Add `in Section \ref{section:CLT}'}\sheilasay{Yes, thanks, DONE} 
 establishes asymptotic normality of the descent statistic on $B_n$ for conjugacy classes with no "short" cycles. 
 Our proof leverages a generating function of Reiner \cite[Theorem 4.1]{ReinerEuropJC1993-2} for the joint distribution of descent and major index by cycle type, an analogue of the corresponding generating function for the symmetric group \cite{GesselReutenauer}.
 The arguments then follow Fulman's analogous result  for descents on conjugacy classes of $\mathfrak{S}_n$ \cite[Theorem 1 and proof of Theorem 2]{FulmanJCTA1998}, but the technical details are nontrivial and require care to execute.
\end{itemize}

\begin{remark}
One essential insight in our work was in developing the notion of \emph{colored permutation constraints} (see Definition~\ref{def:ColoredPermutationConstraints}). It took considerable effort to arrive at this definition, and we discuss these technical difficulties in the full version \cite[Remark~3.3]{GRWCColoredPermutationStatistics}. The fact that Theorem~\ref{thm:IndependenceHigherMoments} and Theorem~\ref{thm:HigherMomentPolynomial} generalize analogous results on the symmetric group \cite{hamaker2022characters, GRWCPermutationStatistics} so cleanly suggests that Definition~\ref{def:ColoredPermutationConstraints} might in fact be the right notion of colored permutation constraints.
 \end{remark}


\section{Preliminaries}\label{sec:preliminaries}

We recall preliminary notions of colored permutation groups. The \emph{colored permutation group} $\mathfrak{S}_{n,r}$ is the wreath product \cite[Chapter 4]{JamesKerber1981}  $\mathbb{Z}_r\wr \mathfrak{S}_n$, where $\mathfrak{S}_n$ is the symmetric group on $n$ elements and $\mathbb{Z}_r$ is the cyclic group on $r$ elements. A \emph{colored permutation} $(\omega, \tau)\in \mathfrak{S}_{n,r}$ can be expressed as an ordered pair consisting of a permutation $\omega \in \mathfrak{S}_n$ along with  a function $\tau:[n]\rightarrow \mathbb{Z}_r$, where the representative elements of $\mathbb{Z}_r$ are taken in $\{0, \dots, r-1\}$.  The value $\tau(j)$ is called the \emph{color} of the symbol $j$, and $\tau(j) + \tau'(j)$ is defined as a sum  of elements in $\mathbb{Z}_{r}$. 

The colored permutation group $\mathfrak{S}_{n,r}$ has a  canonical embedding as a subgroup of the symmetric group $\mathfrak{S}_{rn}$, which we describe explicitly as follows. Writing $[n]^r$ for the set of $rn$ elements $\{i^j | i\in [n], j\in \{0,1,\ldots, r-1\}\}$ where the exponent indicates the color of an element in $[n]$, we can also think of the colored permutation $(\omega, \tau)$ as a bijection $f:[n]^r \rightarrow [n]^r$ defined by  $f(i^j) = \omega(i)^{\tau(\omega(i))+j}$ for all $i,j$, where $\tau(\omega(i))+j$ is taken modulo $r$. In this sense, the coloring of the symbols $\tau$ and the underlying permutation $\omega$ are independently specified.   

We now turn to discussing the conjugacy class structure of $\mathfrak{S}_{n,r}$. An \emph{$r$-partition} of $n\in \mathbb{N}$ is an $r$-tuple of partitions $\rpart=(\lambda^j)_{j=0}^{r-1}$ where each $\lambda^j$ is a partition of some $n_j$ such that $\sum_{j=0}^{r-1} n_j=n$. When $r=2$, we also call this a \emph{bi-partition.} For a cycle in a permutation in $\mathfrak{S}_{n,r}$, the length of this cycle is the number of elements in it, and the color of this cycle is the sum of the colors in the cycle, taken modulo $r$. The \emph{cycle type} of $(\omega,\tau)\in \mathfrak{S}_{n,r}$ is the $r$-partition $\rpart=(\lambda^{j})_{0\leq j\leq r-1}$, where each $\lambda^{j}$ consists of the cycles of color $j$. Then  $m_i(\lambda^{j})$  denotes the number of cycles in $\lambda^{j}$ of length $i$, and $C_{\rpart}$ denotes the elements in $\mathfrak{S}_{n,r}$ with cycle type $\rpart$.

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

\begin{ex}\label{ex:coloredpermutation}
Let $\omega\in \Sym_5$ be the permutation specified by $\omega=[45132]=(143)(25)$ in one-line and  cycle notation.
    Let $\tau = (3,0,1,1,3).$
    The colored permutation   $(\omega,\tau)\in \mathfrak{S}_{5,4}$ is completely specified by the function 
    $f:[5]^4 \rightarrow [5]^4$ satisfying   $f(i^0) = \omega(i)^{\tau(\omega(i))}.$  
     Hence  in two-line, one-line, and cycle notations we have:
    \begin{align*}
        (\omega,\tau)=\left(\begin{matrix}
1^0 & 2^0 & 3^0 & 4^0 & 5^0 \\
4^1 & 5^3 & 1^3 & 3^1 & 2^0 
\end{matrix}\right)=[4^1 5^3 1^3 3^1 2^0]
=(1^3 4^1 3^1)(2^0 5^3).
    \end{align*}
    It has a 3-cycle of color 1 and a 2-cycle of color 3. Its cycle type is thus $(\emptyset, (3), \emptyset, (2))$.
\end{ex}

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

The conjugacy classes of $\mathfrak{S}_{n,r}$ are well understood in terms of cycle type.

\begin{proposition}\cite[Theorem~4.2.8, Lemmas~4.2.9-4.2.10]{JamesKerber1981} \label{cycletype}
    The conjugacy classes of $\mathfrak{S}_{n,r}$ are given  by $C_{\rpart}$, where $\rpart$ is an $r$-partition of $n$.  
\end{proposition}

In the special case $r=2$, the hyperoctahedral group $\mathfrak{S}_{n,2}=B_n$ can be viewed as the group of signed permutations, i.e., bijections on $[\pm n]=\{\pm 1, \pm 2,\ldots, \pm n\}$ where positive and negative elements respectively correspond to colors 0 and 1. In this case, we will denote bipartitions as $(\lambda,\mu)$ and the corresponding conjugacy class as $C_{\lambda,\mu}$. %, i.e. injective maps from $[n]$ to $[\pm n]=\{\pm 1, \pm 2,\ldots, \pm n\}$.
%generated using the elements $s_0=[-1,2,3,\ldots,n]=(-1)$ and $s_i=[1,\ldots,i-1,i+1,i,i+2,\ldots,n]=(i, i+1)$ for $1\leq i\leq n-1$.  With respect to these generators,  %the type $B$ inversion (or length) statistic and 


The type $B$ descent statistic, whose distribution is the subject of Section~\ref{section:CLT}, is then given by the following definition, with the convention that $\omega(0)=0$.  See \cite[Proposition 8.1.2]{BB}:
\begin{equation}\label{eqn:desB}\des_B(\omega)=|\{i\in \{0\}\cup [n-1]\mid \omega(i)>\omega(i+1)\}|.\end{equation}
 

\noindent Two other $B_n$-statistics that will be useful for illustrative purposes are $\inv$ and $\negative$,  defined by (see \cite[Equation 8.1 and page 308]{BB}) 
\begin{equation}\label{eqn:inv-negsum}\inv\!=\!|\{(i,j)\in [n]\times [n] \mid i<j \text{ and }\omega(i)>\omega(j)\}|, \ \ 
\negative(\omega)\!=\!\!\!\! \sum_{i\in [n],\omega(i)<0} \omega(i).\end{equation}
\noindent Also, the Coxeter length statistic $\inv_B$ is given by the formula \cite[Proposition 8.1.1]{BB}
\begin{equation}\label{eqn:InvA-InvB-Neg}\inv_B(\omega)=\inv(\omega)-\negative(\omega)
\end{equation}
We will use the $\des,\inv,$ and $\negative$ statistics as running examples to illustrate our work.  Results on $\inv$ and $\negative$ naturally lead to statements about $\inv_B$, illustrating the more general fact that our results behave nicely with statistics that are defined as linear combinations of other statistics.

Throughout this paper, we will use $\pr_{\mathfrak{S}_{n,r}}$ and $\pr_{\rpart}$ to denote the probabilities in $\mathfrak{S}_{n,r}$ and $C_{\rpart}$ (with respect to the uniform measure). We similarly use $\E_{\mathfrak{S}_{n,r}}$ and $\E_{\rpart}$ for the expected values on the corresponding probability spaces.


\section{Moments of colored permutation statistics}\label{section:colored}

In this section, we will discuss the techniques involved in establishing the independence result, Theorem~\ref{thm:IndependenceHigherMoments}, and the polynomiality result, Theorem~\ref{thm:HigherMomentPolynomial}.

\subsection{Colored permutation constraints}

In this section, we will extend the notion of a permutation constraint from the setting of the symmetric group to the setting of colored permutations. We compare this to \cite[Definition~7.1]{GRWCPermutationStatistics} as well as to the work of Hamaker and Rhoades \cite{hamaker2022characters}, where permutation constraints are called \textit{partial permutations}. A colored permutation constraint will have two components $(K,\kappa)$.  The first,  $K$, will constrain a permutation $\omega$ by specifying a subset of its values. The second component, $\kappa$,  will assign colors to these values.


\begin{definition} \label{def:ColoredPermutationConstraints}
Let $K = \{ (i_{1}, j_{1}), \ldots, (i_{m}, j_{m})\}$ consist of distinct ordered pairs, where $i_{h}, j_{h} \in [n]$. Let $\kappa : \{j_1, \dots, j_m \} \to \mathbb{Z}_{r}$. We call the pair $(K, \kappa)$ a \textit{colored permutation constraint}, and we call $m$ the size of the constraint. 
For $(\omega, \tau) \in \mathfrak{S}_{n,r}$, we  say that $\omega$ satisfies $K$ if $\omega(i_{h}) = j_{h}$ for all $h \in [m]$, and we say $\tau$ satisfies $\kappa$ if $\tau(x) = \kappa(x)$ for all $x \in \{j_1, \dots, j_m\}$. Finally we say that $(\omega, \tau) \in \mathfrak{S}_{n,r}$ satisfies $(K, \kappa)$ if  $\omega$ satisfies $K$ and $\tau$ satisfies $\kappa$. We will sometimes denote a constraint as a set of ordered pairs \[(K,\kappa)=\left\{\left(i_h^0,j_h^{\kappa(j_h)}\right)\right\}_{h=1}^m\]
recording these conditions, and we sometimes omit set braces for brevity.
\end{definition}

Recall from Section~\ref{sec:preliminaries} that we view the hyperoctahedral group $\mathfrak{S}_{n,2}=B_n$ as the group of signed permutations.  In this case, a constraint is of the form  $(K,\kappa)=\left\{\left(i_h,{\kappa(j_h) j_h}\right)\right\}_{h=1}^m $, where $\kappa(j_h)=\pm 1$.




\begin{definition}
    Let $\mathcal{C}$ be a set of colored permutation constraints. The \emph{size} of $\mathcal{C}$ is defined as the maximum size over all constraints contained in $\mathcal{C}$, namely,
    \[\size(\mathcal{C})=\max_{(K,\kappa)\in \mathcal{C}} |K |.\]
    %i.e., the maximum size over all constraints contained in $\mathcal{C}$.
\end{definition}

Recall that a colored permutation statistic is simply a map $X: \mathfrak{S}_{n,r} \rightarrow \mathbb{R}$. We now introduce decompositions of colored permutation statistics as weighted sums of indicator functions corresponding to colored permutation constraints.


\begin{definition}\label{def:weight-color-constraint}
    A colored permutation statistic $X$ is \emph{realizable} over a constraint set of size $m$ if there exists a set of constraints $\mathcal{C}$ of size $m$ and weights $\wt(K,\kappa)\in \mathbb{R}$ such that $X=\sum_{(K,\kappa) \in \mathcal{C}} \wt(K,\kappa)I_{(K,\kappa)}$, where $I_{(K,\kappa)}$ is the indicator function that a permutation satisfies the constraint $(K,\kappa)$.  Note that in general, the decomposition $\sum_{(K,\kappa) \in \mathcal{C}} \wt(K,\kappa)I_{(K,\kappa)}$ is not unique. 
\end{definition}


\begin{ex}\label{ex:des-inv-negsum-constraints}
Many statistics have a natural decomposition in terms of constraints. For the statistics defined on $B_n$ given in Section~\ref{sec:preliminaries}, we have
        \[\des_B =\sum_{j\in [n]} I_{(1,-j)}+\sum_{i\in [n-1]} \,\sum_{\substack{j_1,j_2\in [\pm n]\\ j_1<j_2}} I_{(i,j_2),(i+1,j_1)},\]
        \[\inv = \sum_{\substack{i_1,i_2\in [n] \\ i_1<i_2}} \,\sum_{\substack{j_1,j_2\in [\pm n]\\ j_1<j_2}} I_{(i_1,j_2),(i_2,j_1)},\]
        \[\negative =\sum_{i\in [n]} \, \sum_{j\in [n]} (-j)I_{(i, -j)}.\]
This shows that $\des_B$ and $\inv$ are realizable over constraint sets of size $2$, and $\negative$ is realizable over a constraint set of size $1$. Since $\inv_B$ is the difference of $\inv$ and $\negative$, we also see that $\inv_B$ is realizable over a constraint set of size $2$. 
\end{ex}


\begin{remark}
    We say that $(K, \kappa)$ is \textit{well-defined} if all of the $i_{h} \in [n]$ are distinct, and all of the $j_{h} \in [n]$ are distinct. Observe that if $(K,\kappa)$ is not well-defined, then $I_{(K,\kappa)}$ is identically 
    $0$ on $\mathfrak{S}_{n,r}$, and hence can be omitted from any set realizing a given statistic. Consequently, we are only interested in well-defined constraints.
\end{remark}


\subsection{Independence of moments}\label{sec:Indep-higher-mom}

In this section, we outline the steps leading to the proof of our independence result, Theorem~\ref{thm:IndependenceHigherMoments}. Our methods follow the strategy of \cite[Section 7]{GRWCPermutationStatistics}. Proofs appear in \cite{GRWCColoredPermutationStatistics}.

\begin{definition}\label{def:acyclic}
A colored permutation constraint $(K,\kappa)$ is \emph{acyclic} if $K$ is well-defined and the graph 
 $G(K, \kappa)$, with vertex set $V = [n]$ and directed edge set $K$, does not contain any cycles. Observe that in this case, $G(K, \kappa)$ consists of a set of paths.
\end{definition}



 As a non-example, the size one constraint induced by $I_{(i,-i)}$ from Example~\ref{ex:des-inv-negsum-constraints} is \emph{not} acyclic.
\begin{lemma} \label{lem:HigherMomentsIndependence} (Compare to $\mathfrak{S}_{n}$, cf.  \cite[Lemma 7.15]{GRWCPermutationStatistics})
Consider the group of all $r$-colored permutations $\mathfrak{S}_{n,r}$.  Let $C_{\rpart}$ be a conjugacy class of $\mathfrak{S}_{n,r}$.  Let $(K, \kappa)$ be a well-defined colored permutation constraint of size $m\leq n$, and suppose that each partition in $\rpart$ has all parts of size at least $m+1$.
If $K$ is acyclic, then
\begin{center}
$\pr_{\rpart}[(\omega, \tau) \text{\ satisfies\ } (K, \kappa)] = \frac{1}{(n-1)(n-2) \cdots (n-m)} \cdot \frac{1}{r^{m}}.$
\end{center}
\noindent If $K$ is not acyclic, then $\pr_{\rpart}[(\omega, \tau) \text{\ satisfies\ } (K, \kappa)] = 0.$
\end{lemma}

One essential observation in proving Lemma~\ref{lem:HigherMomentsIndependence} is that the permutation and the coloring can be treated independently. 


Lemma~\ref{lem:HigherMomentsIndependence} can be used to analyze the first moment of a statistic $\E_{\rpart}[X]$ by expressing $X$ in terms of constraints. We need one final lemma to accommodate arbitrary moments 
$\E_{\rpart}[X^k]$ in the main result of this section, Theorem~\ref{thm:IndependenceHigherMoments}.

\begin{lemma} \label{lem:Thm7.16}
Let $X_{1}, X_{2} : \mathfrak{S}_{n,r} \rightarrow \mathbb{R}$ be realizable over constraint sets of size $m_{1}, m_{2}$ respectively. Then $X_1X_2$ is realizable over a constraint set of size $m_1+m_2$. In particular, for any integer $k \geq 1$, we have that $X_{1}^{k}$ is realizable over a constraint set of size $km_1$.
\end{lemma}

This leads to the main theorem of this section. 

\begin{theorem} \label{thm:IndependenceHigherMoments}
    Suppose $X:\mathfrak{S}_{n,r}\to \mathbb{R}$ is realizable over a constraint set of size $m$. For any $k\geq 1$, the $k$th moment $\E_{\rpart}[X^k]$ coincides on all conjugacy classes $C_{\rpart}$ with no cycles of length $1,2,\ldots,mk$.
\end{theorem}

Note that the above theorem makes precise the notion of ``short" cycles. In particular, if we are considering the $k$th moment of a statistic $X$ realizable over a constraint set of size $m$, then the ``short" cycles are the ones of length at most $mk$.


\begin{remark}
    Note that a colored permutation $(\omega,\tau)$ is itself a colored permutation constraint of size $n$. Hence, we can express any statistic $X$ using size $n$ constraints. Additionally, one can show that if $X$ is realizable over a constraint set of size $m$, then it is also realizable over a constraint set of size $m'$ for $m\leq m'\leq n$. For the full strength of our results, we are primarily interested in minimizing $m$, and we call this minimum possible value the \emph{size} of $X$.
\end{remark}


\begin{remark}\label{rem:negsum}
    The arguments leading to the proof of  Theorem \ref{thm:IndependenceHigherMoments} have practical applications for computing moments of statistics on those conjugacy classes. For example,  consider $\negative$ on $B_n$, which can be expressed as
    $\negative = \sum_{i\in [n]} \, \sum_{j\in [n]} (-j)I_{(i, -j)}.$
    Note that here all constraints are acyclic except for $(i,-i)$. 
    One can then show that  for any bi-partition $(\lambda,\mu)$ of $n$ where all the parts 
    %of each partition 
    have size at least $2$, 
    \begin{equation*}
        \begin{split}
            \E_{\lambda,\mu}[\negative] & = -\sum_{i\in [n]} i\cdot \E_{\lambda,\mu}[I_{(i,-i)}]- \sum_{i\in [n]} \sum_{j\in [n]\setminus i}j\cdot  \E_{\lambda,\mu}[I_{(i,-j)}]\\
            & = -\frac{1}{(n-1)\cdot 2}\cdot \sum_{i\in [n]} \, \sum_{j\in [n]\setminus i} j
            =-\frac{1}{2}\binom{n+1}{2}.
        \end{split}
    \end{equation*}
    More generally, one can use $\negative=\sum_{i\in [n]} \, \sum_{j\in [n]} (-j)I_{(i, -j)}$ to express $\negative^k$ using constraints of size at most $k$. On conjugacy classes where all parts have size at least $k+1$, a similar approach as the one above can be used to calculate $\E_{\lambda,\mu}[\negative^k]$.
\end{remark}


\subsection{Symmetric colored permutation statistics}\label{sec:symmetric-colored-perms}

We now turn to extending the notion of a \textit{symmetric} permutation statistic from \cite{GRWCPermutationStatistics} to the colored setting. We begin with some definitions. 

\begin{definition}
The \textit{support} of a colored permutation constraint $(K, \kappa) = \{(i_r^0,j_r^{\kappa(j_r)})\}_{r=1}^m$ is $\text{supp}(K, \kappa) = \{ i_{1}, \ldots, i_{m}, j_{1}, \ldots, j_{m}\}$. We emphasize that $\supp(K, \kappa)$ is a set and not a multiset. 
\end{definition}

\begin{definition}
    Consider any colored permutation constraint $(K,\kappa)$ with support given by $a_{1} < \cdots < a_{s}$. For any order-preserving injection $f : \{ a_{1}, \ldots, a_{s}\} \to [n]$, define $f(K,\kappa)$ to be the constraint \[f(K,\kappa)=(\,\{(f(i_1),f(j_1)),\ldots,(f(i_m),f(j_m))\},\,\{\kappa(f(j_1))=k_1,\ldots, \kappa(f(j_m))=k_m\}\,).\]
\end{definition}

\begin{definition}\label{def:symmetric-statistic}
    A set of colored permutation constraints $\mathcal{C}$ is \emph{symmetric} if for all $(K,\kappa)\in \mathcal{C}$ and any order-preserving injection $f:\operatorname{supp}(K,\kappa)\to [n]$, we have $f(K,\kappa)\in \mathcal{C}$. A statistic $X$ is \emph{symmetric} if it has the form $X=\sum_{(K,\kappa)\in \mathcal{C}} I_{(K,\kappa)}$ for some symmetric $\mathcal{C}$.
\end{definition} 


%
Many statistics naturally satisfy this condition.
\begin{ex} 
Consider the statistic $\inv$ on $B_n$ that can be realized as
    \[\inv=\sum_{ \substack{i,j \in [n]\\ i<j}} \, \sum_{\substack{k,\ell\in [\pm n]\\ k<\ell} }I_{\{(i,\ell),(j,k)\}}.\]
     We denote the constraint set $\mathcal{C}$. If $k,\ell>0$, then for any order preserving $f:\{i,j,k,\ell\}\to [n]$, we see that $\{(f(i),f(\ell)),(f(j),f(k))\}\in \mathcal{C}$. Note that the set $\{i,j,k,\ell\}$ need not consist of four distinct elements. If $k<0$ and $\ell>0$, we see that for any order-preserving $f:\{i,j,|k|,\ell\}\to [n]$, we have $\{(f(i),f(\ell)),(f(j),-f(|k|))\}\in \mathcal{C}$. The same argument holds for the case when $k,\ell<0$. 
\end{ex}


\begin{definition} \label{def:SymmetricExtensions}
    Fix $n_0\ge 2$. Let $X=\sum_{(K,\kappa)\in \mathcal{C}} I_{(K,\kappa)}$ be a symmetric statistic defined on $\mathfrak{S}_{n_0,r}$. Define the \emph{$r$-colored symmetric extensions} of $X$ to be the statistics  $X_n=\sum_{(K,\kappa)\in \mathcal{C}_n} I_{(K,\kappa)}$ on $\mathfrak{S}_{n,r}$ with $\mathcal{C}_n$ defined as follows:
    \begin{itemize}
        \item If $n\leq n_0$, then $\mathcal{C}_n$ contains all $(K,\kappa)\in \mathcal{C}$ with support contained in $[n]$.
        \item If $n\geq n_0$, then $\mathcal{C}_n$ is the set of all $f(K,\kappa)$ where $(K,\kappa)\in \mathcal{C}$ and  $f:[n_0]\to [n]$ is order-preserving.
    \end{itemize}
    Observe that by construction, each $X_n$ is a symmetric statistic. We emphasize here that $r$ is kept constant throughout this construction. 
\end{definition}

Many statistics can be constructed in this manner. For example, if $\mathcal{C}$ is the set of constraints for $\inv$ on $B_4$, then this results in the $\inv$ statistics on all $B_n$. In general, the moments of these statistics satisfy the following polynomial property. 


\begin{theorem} \label{thm:HigherMomentPolynomial}
Fix $r\ge 1$. Let $(X_{n})$ be the symmetric extensions of a symmetric statistic $X=X_{n_0}$ on $\mathfrak{S}_{n,r}$ induced by a constraint set $\mathcal{C}$ of size $m$. There exists a polynomial $p_X(n)$ of degree at most $mk$ depending only on $X$ such that $p_X(n) = \E_{\rpart_{n}}[X_{n}^k]$ for any $r$-partition $\rpart_n$ of $n$ where all $\lambda_n^{(j)}$ have parts of size at least $mk+1$. 
\end{theorem}

Note that one can show this polynomiality property for other statistics that are not symmetric extensions. The key requirement is that the weights for the various $I_K$ behave in a way that allows us to divide by the denominators that result from applying \cref{lem:HigherMomentsIndependence}.

\section{Descents in conjugacy classes of hyperoctahedral groups}\label{section:CLT}


In this section we discuss the techniques involved in establishing our central limit theorem for descents in conjugacy classes of $B_{n}$ that do not have short cycles. The descent statistic on $B_n$ was defined in Eqn.~\eqref{eqn:desB}.
Let $(\lambda(\omega),\mu(\omega))$ denote  the cycle type of $\omega\in B_n$, and let $m_i(\lambda)$ denote the number of parts of $\lambda$ equal to $i$.
While Reiner \cite{ReinerEuropJC1993-2} uses a different notion of descents, the generating function \cite[Theorem 5.3]{FulmanKimLeePetersen2021}
\begin{equation}\label{eq:des_gen_function}
    \sum_{\omega \in B_n} t^{\des_B(\omega)} \prod_i x_i^{m_i(\lambda(\omega))} y_i^{m_i(\mu(\omega))}
\end{equation}
    is unaffected.



Following Fulman \cite{FulmanJCTA1998}, our approach involves examining the generating function given in (\ref{eq:des_gen_function}),
which allows us to analyze the generating function for $\des_B$ on a conjugacy class. We then relate this with the generating function for descents on all of $B_n$. In the case where there are no short cycles in $C_{\lambda,\mu}$, we will ultimately conclude that certain moments of $\des_B$ agree on $C_{\lambda,\mu}$ and $B_n$, and this in turn enables us to use the method of moments with a known central limit theorem of Chow and Mansour for $\des_B$ on $B_n$ given below.


\begin{proposition}\cite[Thm 3.4]{CM2012}\label{prop:norm-allBn} Let $X_n$ be $\des_B$ defined on $B_n$. Then $X_n$ has mean $n/2$ and variance $(n+1)/12$, and as $n\to\infty$, the standardized random variable $(X_n-n/2)/\sqrt{(n+1)/12}$ converges to a standard normal distribution.
    \end{proposition}


   We will need the well-known generating function of $\des_B$ over all of $B_n$.


     \begin{proposition}\cite[Eqn.~(13.3)]{Petersen2015}\label{prop:Bn_genfunction} Let ${B}_n(t)=\sum_{\omega \in B_n} t^{\des_B(\omega)+1}$. 
    Then 
    \begin{equation*}\frac{{B}_n(t)}{(1-t)^{n+1}}=\sum_{k \geq 1} (2k-1)^n t^k.\end{equation*}
    \end{proposition}


We now analyze (\ref{eq:des_gen_function}), which will allow us to derive an expression for the generating function of $\des_B$ on a conjugacy class $C_{\lambda,\mu}$. 
The following  expression features prominently  in our analysis.


\begin{definition}\cite{ReinerEuropJC1993-2}\label{def:exponent-Reinernecklaces}
Let  $\mu(d)$ be the number-theoretic M\"{o}bius function.  Define, for nonnegative 
 integers $r$ and $m$, \begin{equation*}\label{eq:mobius}
N(r, 2m)=\frac{1}{2m} \sum_{\substack{d\vert m \\ d \text{ odd}}} \mu(d) \left(r^{m/d}-1\right).
\end{equation*} 
\end{definition}

Reiner \cite[Theorem 4.1, Theorem 4.2]{ReinerEuropJC1993-2} shows that $N(2k-1, 2m)$ must be a nonnegative integer for all $k,m \ge 1$.

For a fixed bi-partition $(\lambda,\mu)$ of $n$, we use the special case of \cite[Theorem 4.1]{ReinerEuropJC1993-2} appearing in \cite[Theorem 5.3]{FulmanKimLeePetersen2021}  to derive the following expressions for the generating function $B_{\lambda,\mu}(t)$  $=\sum_{\omega \in C_{\lambda,\mu}} t^{\des_B(\omega)+1}$ of descents over the conjugacy class $C_{\lambda,\mu}$.

\begin{proposition}\label{prop:all}
Let 
$\lambda=(1^{{m_1(\lambda)}},2^{{m_2(\lambda)}},\ldots)$ and $\mu=(1^{{m_1(\mu)}},2^{{m_2(\mu)}},\ldots)$. Then the following are equal to $B_{\lambda,\mu}(t)/(1-t)^{n+1}$:
\begin{eqnarray*} 
& t \delta_{((1^n), \emptyset)}+\sum_{k \geq 2} t^k  \left(\prod_{i\ge 1} \binom{N(2k-1, 2i)}{m_i(\mu)} \,
\prod_{i\ge 2} \binom{N(2k-1, 2i)+m_i(\lambda)-1}{m_i(\lambda)}\right)\binom{N(2k-1, 2)+m_1(\lambda)}{m_1(\lambda)} \label{eq:conjugacy1}\\
&=t \delta_{((1^n), \emptyset)}+\sum_{k \geq 2} t^k \frac{m_1(\lambda)+k-1}{k-1} \prod_{i \geq 1} \binom{N(2k-1, 2i)-1+m_i(\lambda)}{m_i(\lambda)} \binom{N(2k-1, 2i)}{m_i(\mu)}\label{eq:conjugacy2} .
\end{eqnarray*}
Here $\delta_{((1^n), \emptyset)}$ is the Kronecker delta which is 1 for the conjugacy class  $\lambda,\mu=((1^n), \emptyset)$, and zero otherwise.
\end{proposition}

By solving for $B_{\lambda,\mu}$ and extracting the coefficient of $t^d$, we also obtain the following corollary.

\begin{corollary}
The number of permutations $\omega \in B_n$ that are of cycle type $(\lambda,\mu)$ and have $d-1$ descents is
\begin{eqnarray*}
& \sum_{k=1}^d (-1)^{d-k} \binom{n+1}{d-k} \binom{m_1(\lambda)+k-1}{m_1(\lambda)} \prod_{i\ge 2} \binom{N(2k-1, 2i)+m_i(\lambda)-1}{m_i(\lambda)} \prod_{i\ge 1} \binom{N(2k-1, 2i)}{m_i(\mu)}.
\end{eqnarray*}
\end{corollary}


We now give an elegant analogue of a result of Fulman \cite[Proof of Theorem 2]{FulmanJCTA1998}, which will relate $B_{\lambda,\mu}(t)$ and $B_n(t)$. 

\begin{theorem}\label{thm:expansion}
Let $C_{\lambda,\mu}$ be the conjugacy class of $B_n$ indexed by the bi-partition $(\lambda,\mu)$ of $n$, let $B_n(t)=\sum_{\omega \in B_n} t^{\des_B(\omega)+1}$, and let $B_{\lambda,\mu}(t)=\sum_{\omega \in C_{\lambda,\mu}} t^{\des_B(\omega)+1}$. Then
\begin{eqnarray*}\label{eq:B}
\frac{B_{\lambda,\mu}(t)}{|C_{\lambda,\mu}|}=\frac{B_n(t)}{2^n n!}+\frac{1-t}{2n} \frac{B_{n-1}(t)}{2^{n-1} (n-1)!} [m_1(\lambda)^2-m_1(\mu)^2]+(1-t)^2 g(t),
\end{eqnarray*}
where $g(t)$ is some polynomial in $t$. Furthermore, when all cycles in $C_{\lambda,\mu}$ have length larger than $2k$, 
\begin{eqnarray*}
\frac{B_{\lambda,\mu}(t)}{|C_{\lambda,\mu}|}=\frac{B_n(t)}{2^n n!}+(1-t)^{k+1} h(t),
\end{eqnarray*}
where $h(t)$ is some polynomial in $t$.
\end{theorem}

The latter case allows us to obtain the following result involving moments of $\des_B$ on $B_n$ and $C_{\lambda,\mu}$. 

\begin{corollary}\label{cor:Bn}
Let $C_{\lambda,\mu}$ be the conjugacy class of $B_n$ indexed by the bi-partition $(\lambda,\mu)$ of $n$. The $k$th moment of $\des_B$ in $C_{\lambda,\mu}$ is equal to the $k$th moment of $\des_B$ in $B_n$ if all cycles in $C_{\lambda,\mu}$ have length greater than $2k$. 
\end{corollary}

The main result of this section, Theorem~\ref{thm:CLTMainS}, now follows by applying Corollary~\ref{cor:Bn}, the method of moments, and the asymptotic normality theorem for descents in $B_n$ given in Proposition~\ref{prop:norm-allBn}.

\begin{theorem} \label{thm:CLTMainS}
For every $n\geq 1$, pick a conjugacy class $C_{\lambda_n, \mu_n}$ in $B_n$ indexed by the bi-partition $(\lambda_n,\mu_n)$ of $n$, where $\lambda_n=(1^{{m_1(\lambda_n)}},2^{{m_2(\lambda_n)}},\ldots)$ and $\mu_n=(1^{{m_1(\mu_n)}},2^{{m_2(\mu_n)}},\ldots)$. Define $X_n$ to be $\des_B$ on $C_{\lambda_n,\mu_n}$. Suppose that for all $i$, $m_i(\lambda_n) \rightarrow 0$ and $m_i(\mu_n) \rightarrow 0$ as $n \rightarrow \infty$. For sufficiently large $n$, $X_n$ has mean $n/2$ and variance $(n+1)/12$. Furthermore, as $n\to\infty$, the random variable $(X_n-n/2)/\sqrt{(n+1)/12}$ converges to a standard normal distribution.
\end{theorem}


\section{Conclusion}
In this paper, we have introduced a notion of constraints and size for any colored permutation statistic $X:\mathfrak{S}_{n,r}\to \mathbb{R}$, and we have used this framework to study the moments of $X$ on conjugacy classes $C_{\rpart}$. In particular, we have established that for a statistic of size $m$,  the $k$th moment on $C_{\rpart}$ is independent of conjugacy class $C_{\rpart}$ when all parts of the partitions in $\rpart$ have length at least $mk+1$. For statistics on $\mathfrak{S}_{n,r}$ that can be expressed as symmetric extensions, these moments are polynomials in $n$. Our results directly generalize those in \cite{GRWCPermutationStatistics} on $\mathfrak{S}_n$. Given the numerous connections to \cite{hamaker2022characters}, one natural problem is the following. 

\begin{problem}
    Use the representation theory of $B_n$ and $\mathfrak{S}_{n,r}$ to establish analogues of the results in \cite{hamaker2022characters}.
\end{problem}



Finally, we note that $\mathfrak{S}_n$ and $B_n$ are respectively the type $A$ and type $B$ Coxeter groups. The following is a natural problem to consider next. 

\begin{problem}
    Establish analogues of the results in this paper for the type $D$ Coxeter groups.
\end{problem}

It would also be of interest to establish analogous results for (irreducible) complex reflection groups.


\acknowledgements{This work began at the 2022 Graduate Research Workshop in Combinatorics, which was supported in part by NSF grant \#1953985 and a generous award from the Combinatorics Foundation. We thank Yan Zhuang for alerting us to the work of Hamaker and Rhoades \cite{hamaker2022characters}, and Zach Hamaker for helpful discussions regarding connections to \cite{hamaker2022characters}. We would also like to thank the anonymous referees for helpful feedback.} 

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