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\title
[\texorpdfstring{$q$}{q}-analogue of C-S-V result]
{A \texorpdfstring{$q$}{q}-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets}
\author[\initial{Y.} Li]{\firstname{Yifei} \lastname{Li}}
\address{University of Illinois at Springfield\\
Department of Mathematical Sciences and Philosophy\\
One University Plaza\\
MS WUIS 13\\
Springfield\\
Illinois 62703 (USA)}
\email{yli236@uis.edu}
\thanks{The author would like to thank John Shareshian for his insightful comments and valuable advice.}
\keywords{algebraic combinatorics, poset homology, shellability, symmetric functions, symmetric group representation}
\subjclass{05E05, 05E10, 05E18, 05E99, 20C30}
\begin{document}
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\begin{abstract}
Let $f(z)=\sum_{n=0}^{\infty}(-1)^nz^n/n!n!$. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series $1/f(z)=\sum_{n=0}^{\infty}\omega_n z^n/n!n!$. They proved that $\omega_n$ counts the number of pairs of permutations of the $n$th symmetric group $\mathcal{S}_n$ with no common ascent. This paper gives a combinatorial interpretation of a natural $q$-analogue of $\omega_n$ by studying the top homology of the Segre product of the subspace lattice $B_n(q)$ with itself. We also derive an equation that is analogous to a well-known symmetric function identity: $\sum_{i=0}^n(-1)^ie_ih_{n-i}=0$, which then generalizes our $q$-analogue to a symmetric group representation result.
\end{abstract}
\maketitle
\section{Introduction} \label{Intro}
Consider the power series $f(z)=\sum_{n=0}^{\infty} (-1)^n\frac{z^n}{n!n!}$ and define the numbers $\omega_0,\omega_1,$ $\omega_2,\dots$ by $\frac{1}{f(z)}=\sum_{n=0}^{\infty}\omega_n\frac{z^n}{n!n!}$. It follows quickly from the definition that for $n\geq 1$,
\begin{equation} \label{CSV_intro}
\sum_{k=0}^n(-1)^k \binom{n}{k}^2 \omega_k=0.
\end{equation}
Given $\sigma\in \mathcal S_n$, a permutation of $[n]\coloneqq \{1,2,\dots,n\}$, we call $i\in [n-1]$ an \emph{ascent} of $\sigma$ if $\sigma (i)<\sigma (i+1)$.
Carlitz, Scoville and Vaughan proved the following result:
\begin{thm} \label{CSV_Othm} \emph{(Carlitz, Scoville, and Vaughan~\cite{CSV})}
The number $\omega_k$ in equation \eqref{CSV_intro} is the number of pairs of permutations of $\mathcal S_k$ with no common ascent.
\end{thm}
Two permutations have no common ascent if they do not rise at the same position when written in one-line notation. For example, in one-line notation $(12,21)$, $(21,12)$, $(21,21)$ are all the pairs of permutations of $\{1,2\}$ with no common ascent, so we have $\omega_2=3$. Since the Bessel function $J_0(z)$ is essentially $f(z^2)$, Carlitz, Scoville and Vaughan's result provided a combinatorial interpretation of the coefficient $\omega_k$ in the reciprocal Bessel function. \\
In this paper, we will develop a $q$-analogue of Theorem~\ref{CSV_Othm}. To that purpose, recall that $[n]_q\coloneqq q^{n-1}+q^{n-2}+\cdots+1$ is the $q$-analogue of the natural number $n$ and that $\qbinom{n}{k}_q\coloneqq \frac{[n]_q!}{[k]_q![n-k]_q!}$ is the $q$-analogue of the binomial coefficient $\binom{n}{k}$, where $[n]_q!\coloneqq \prod_{i=1}^{n}{[i]_q}$. For a permutation $\sigma\in \mathcal S_n$, the \emph{inversion statistic} is defined by $$\inv(\sigma)\coloneqq \mid \{(i,j):1\leq i\sigma(j)\}\mid .$$
\begin{thm} \label{Q-CSV} Let $\mathcal D_n$ denote the set $\{(\sigma,\tau)\in \mathcal S_n\times\mathcal S_n \mbox{ }\mid \mbox{ }\sigma$, $\tau$ have no common ascent$\}$, and let $W_n(q)=\sum_{(\sigma,\tau)\in \mathcal D_n}{q^{\inv(\sigma)+\inv(\tau)}}$. Then for $n\geq 1$,
\begin{equation}\label{W_n(q)_intro}
\sum_{i=0}^{n}{\qbinom{n}{i}_q^2 (-1)^i W_i(q)}=0.
\end{equation}
\end{thm}
Put $F(z)=\sum_{n=0}^{\infty} (-1)^n\frac{z^n}{[n]_q![n]_q!}$. The function $F\Big((\frac{z}{2(1-q)})^2\Big)$ is the $q$-Bessel function $J_0^{(1)}(z;q)$. The $q$-Bessel functions were first introduced by F. H. Jackson in 1905 and can be found in later literature (see Gasper and Rahman~\cite{Gasper}). It follows from equation \eqref{W_n(q)_intro} that $\frac{1}{F(z)}= \sum_{n=0}^{\infty}W_n(q)\frac{z^n}{[n]_q![n]_q!}$, giving the coefficients of the reciprocal $q$-Bessel function a combinatorial meaning.
In Section~\ref{section_q-CSV}, we will prove Theorem~\ref{Q-CSV} by studying the top homology of the Segre product of the subspace lattice $B_n(q)$ with itself. From a poset homology perspective, the coefficient $W_n(q)$ is a signless Euler characteristic and counts the number of decreasing maximal chains of this Segre product poset. All definitions will be reviewed in this section.
In Section~\ref{section_xch_map}, we define the product Frobenius characteristic map to serve as a useful tool in studying representations of the product group $\mathcal S_n\times\mathcal S_n$. We then further generalize our $q$-analogue to a symmetric group representation result in Section~\ref{section_sym_analogue} (see Theorem \ref {Stanley q-form}) using the Whitney homology technique. This generalization is an analogue of the well-known symmetric function identity: $\sum_{i=0}^n(-1)^ie_ih_{n-i}=0$.
Finally, in Section~\ref{alt_proof_CSV} we point out that an alternative proof of Theorem~\ref{CSV_Othm} can be obtained by specializing our proof of Theorem~\ref{Q-CSV} at $q=1$.
\section{The \texorpdfstring{$q$}{q}-analogue of a result of Carlitz, Scoville, and Vaughan} \label{section_q-CSV}
We recall the definition of $B_n(q)$, which is a $q$-analogue of the subset lattice $B_n$. Let~$q$ be a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. Consider the $n$-dimensional linear vector space $\mathbb{F}_q^n$ and its subspaces. Then $B_n(q)$ is the lattice of those subspaces ordered by inclusion. The poset $B_n(q)$ is a geometric lattice, so every element is a join of atoms (\cite{ec1}*{Example 3.10.2}). The poset $B_n(q)$ is graded with a rank function $\rho(W)\coloneqq $ the dimension of the subspace $W$, where a poset is said to be graded if it is pure and bounded.
An \emph{edge labeling} of a bounded poset $P$ is a map $\lambda: \mathcal E(P)\to \Lambda$, where $\mathcal E(P)$ is the set of covering relations $x\coveredby y$ of $P$ and $\Lambda$ is some poset. If $P$ is a poset with an edge labeling $\lambda$, then a maximal chain $c=(\hat{0}\coveredby x_1\coveredby \cdots \coveredby x_t\coveredby \hat{1})$ of $P$ is \emph{increasing} if $\lambda (\hat{0},x_1)<\lambda(x_1,x_2)<\cdots<\lambda(x_t,\hat{1})$. We call the chain $c$ \emph{decreasing} if there is no~$i\in \{1,2,\dots,t\}$ such that $\lambda(x_{i-1},x_i)<\lambda(x_i,x_{i+1})$ in~$\Lambda$. For a chain $c$, we associate a word $$\lambda (c)=\lambda (\hat{0},x_1)\lambda(x_1,x_2)\cdots\lambda(x_t,\hat{1}).$$ If $\lambda (c_1)$ lexicographically precedes $\lambda (c_2)$, we say that $c_1$ lexicographically precedes $c_2$ and we denote this by $c_1<_Lc_2$.
\begin{dfn}[{Bj\"orner and Wachs~\cite{BW2}*{Definition 2.1}}] \label{EL} An edge labeling is called an \emph{EL-labeling (edge lexicographical labeling)} if for every interval $[x, y]$ in $P$,
\begin{enumerate}
\item there is a unique increasing maximal chain $c$ in $[x, y]$, and
\item $c<_Lc'$ for all other maximal chains $c'$ in $[x, y]$.
\end{enumerate}
\end{dfn}
A bounded poset that admits an EL-labeling is said to be \emph{EL-shellable}. We only need to consider pure shellability in this paper since both $B_n(q)$ and the Segre product of $B_n(q)$ with itself (see Definition~\ref{Segre}) are pure and bounded. It is well known that $B_n(q)$ is EL-shellable (see~\cite{Wachs_notes}*{Exercise 3.4.7}) and a general edge-labeling for semimodular lattices is given in~\cite{ec1}. Here we define a specific EL-labeling of $B_n(q)$, which will be used to prove our results. Let $A$ be the set of all atoms of $B_n(q)$. For a subspace of $\mathbb{F}_q^n$, $X\in B_n(q)$, we define $A(X)\coloneqq \{V\in A\mid V\leq X\}$. The following two steps define an edge-labeling on the graded poset $B_n(q)$.
\textbf{1.} For a $1$-dimensional subspace $V$ of $\mathbb{F}_q^n$ (an atom of $B_n(q)$), let $v$ be a basis element of $V$. We define a map
$f$: $A \to [n]$, $f(V)=$ the index of the right-most non-zero coordinate of $v$. For example, in $B_3(3)$, if $V_1=\spn\{\langle 1,0,1\rangle\}$ and $V_2=\spn\{\langle 2,1,0\rangle\}$, $f(V_1)=3$ and $f(V_2)=2$.
\textbf{2}. In the case of $B_3(3)$, if $X=\spn\{\langle 1,0,1\rangle,\langle 2,1,0\rangle\}$, then $A(X)$ also contains $\spn\{\langle 0,1,1\rangle\}$ and $\spn\{\langle 2,2,1\rangle\}$. But any vector whose right-most non-zero coordinate is the first coordinate will not be in $X$. So $f(A(X))=\{2,3\}$ and $\mid f(A(X))\mid =2$. For a $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, Gaussian elimination implies the existence of a basis of $X$ whose elements have distinct right-most non-zero coordinates and this in turn implies that $f(A(X))$ has $\dim(X)$ elements. Let $Y$ be an element of $B_n(q)$ that covers $X$, then $\dim(Y)=\dim(X)+1$. The set $f(A(Y))$\textbackslash $f(A(X))$ is a subset of $[n]$ and has exactly one element. This element will be the label of the edge $(X,\,Y)$.
\begin{prop} \label{B_n(q)_EL} The edge labeling described above is an EL-labeling on the subspace lattice $B_n(q)$.\end{prop}
\begin{proof} Edges in the same chain cannot take duplicate labels since $\mathbb F_q^n$ is $n$-dimensional and any maximal chain must take all labels in $\{1,2,\dots,n\}$. Let~$[X, Y]$ be a closed interval in $B_n(q)$. All maximal chains of $[X,\, Y]$ will take labels from the set~$f(A(Y))$\textbackslash $f(A(X))$. Let $a_1