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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 $$\inv(\sigma)\coloneqq \mid \{(i,j):1\leq i<j\leq n \mbox{ and } \sigma(i)>\sigma(j)\}\mid .$$ 

\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 $a_1<a_2<\cdots <a_l$ be all the elements of~$f(A(Y))$\textbackslash $f(A(X))$ arranged in increasing order. Given $a_i\in f(A(Y))$\textbackslash $f(A(X))$, there exists an atom~$V_i\in A(Y)$ with $f(V_i)=a_i$. Note that $V_i$ is a $1$-dimensional subspace of~$\mathbb{F}_q^n$ and the join of $V_i$ and $X$ is in $[X, Y]$. We build a chain according to the increasing order of $a_i$'s, each time adjoining one $1$-dimensional subspace. Then the chain~$c=(X\coveredby X\vee V_1\coveredby \cdots \coveredby X\vee V_1\vee V_2\vee \cdots \vee V_l=Y)$ is an increasing maximal chain of $[X, Y]$. 

For the uniqueness of the increasing maximal chain, it suffices to show the uniqueness of the selection of $X\vee V_1$ since $X$ and $Y$ are arbitrary. Take $V_1'\in A(Y)$ with $f(V_1')=a_1$. We can find a basis vector $v_1$ of $V_1$ and a basis vector $v_1'$ of $V_1'$ so that the $a_1$th coordinate of both vectors are $1$. Then $v_1-v_1'\in Y$ and $f(\spn\{v_1-v_1'\})<a_1$. Since $a_1<a_2<\cdots<a_l$ are all the elements of $f(A(Y))$\textbackslash $f(A(X))$, $v_1-v_1'$ must be a vector in $X$. Then $X\vee V_1=X\vee V_1'$. Therefore $X\vee V_1$ is unique. At each step of building the increasing maximal chain, there is a unique subspace that gives the connecting edge the smallest label.

Labels of all other maximal chains of $[X,Y]$ are permutations of elements in~$f(A(Y))$\textbackslash $f(A(X))$, which are all lexicographically larger than the label of the unique increasing chain $c=(X\coveredby X\vee V_1\coveredby \cdots \coveredby X\vee V_1\vee V_2\vee \cdots \vee V_l=Y)$. Condition $(2)$ of Definition~\ref{EL} is also satisfied.
\end{proof}


Under this EL-labeling, to each maximal chain of the subspace lattice $B_n(q)$, one can assign a permutation $\sigma$ of $\mathcal{S}_n$. See Section~\ref{Intro} for the definition of the inversion statistic~$\inv(\sigma)$.  

\begin{lem} \label{Chains same perm}
The number of maximal chains of $B_n(q)$ assigned label $\sigma \in \mathcal{S}_n$ is $q^{\inv(\sigma)}$.
\end{lem}

\begin{proof} 
For each $1$-dimensional subspace of $\mathbb{F}_q^n$, we can pick a basis vector that has $1$ on its right-most non-zero coordinate. Given $\sigma \in \mathcal S_n$, for each $i\in [n-1]$, let $\inv_{\sigma(i)}$ denote the number of $j$ such that $1\leq i< j\leq n$ and $\sigma (j)<\sigma(i)$. The number of ways to choose an atom $W_1$ such that the edge $(0,W_1)$ takes label $\sigma(1)$ is clearly $q^{\sigma(1)-1}=q^{\inv_{\sigma(1)}}$. 

Let $k\in [n]$, assume the chain $0\coveredby W_1\coveredby ...\coveredby W_{k-1}$ has label $\sigma (1) \sigma (2)...\sigma (k-1)$. We need to choose a $W_k=W_{k-1}\vee V_k$ so that the edge $(W_{k-1}, W_k)$ takes the label $\sigma(k)$ and $V_k$ is an atom. Pick a basis vector for $V_k$, call it $v_k$, that has $1$ on the $\sigma(k)$th coordinate and all $0$'s after the $\sigma(k)$th coordinate. For all $j$ such that $1\leq k< j\leq n$ and $\sigma (j)<\sigma(k)$, $W_{k-1}$ contains no vector whose right-most non-zero coordinate is the $\sigma(j)$th. Thus, any variation of the values on those $\sigma(j)$th coordinates of $v_k$ will result in a different $W_k$. Then there are $q^{\inv_{\sigma(k)}}$ ways to choose a $W_k$. Therefore, the number of maximal chains assigned label $\sigma$ is $\prod^{i=n}_{i=1}{q^{\inv_{\sigma(i)}}}=q^{\sum^{i=n}_{i=1}{\inv_{\sigma(i)}}}=q^{\inv(\sigma)}$.
\end{proof}

Let us review a simplified definition of the Segre product poset. A general definition can be found in~\cite{Segre_rees}.

\begin{dfn} \label{Segre}  
Let $P$ be a graded poset with a rank function $\rho$. Then the \emph{Segre product poset} of $P$ with itself, denoted by $P\circ P$, is defined to be the induced subposet of the product poset $P\times P$ consisting of the pairs $(x,y)\in P\times P$ such that $\rho(x)=\rho(y).$  
\end{dfn}

Now consider the Segre product of $B_n(q)$ with itself. Using the EL-labeling of $B_n(q)$ described right after Definition~\ref{EL}, the Segre product poset $B_n(q)\circ B_n(q)$ admits the following edge labeling. Given two elements $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ in~$B_n(q)\circ B_n(q)$ satisfying the covering relation $X\coveredby Y$, we must have $X_1\coveredby Y_1$ and $X_2\coveredby Y_2$ in $B_n(q)$. In the EL-labeling of $B_n(q)$, suppose the edge connecting $X_1$ and~$Y_1$ admits a label $i$ and the edge connecting $X_2$ and $Y_2$ admits a label $j$, then the edge connecting $X$ and $Y$ in $B_n(q)\circ B_n(q)$ is labeled by $(i,j)$. 

\begin{cor} \label{Segre_EL} \emph{(of Proposition~\ref{B_n(q)_EL})}
The edge-labeling of $B_n(q)\circ B_n(q)$ defined above is an EL-labeling.
\end{cor}

\begin{proof}
Let $[X,Y]$ be any closed interval in $B_n(q)\circ B_n(q)$. The elements $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$, where $X_1\leq Y_1$ and $X_2\leq Y_2$ in $B_n(q)$. So $[X_1,Y_1]$ and $[X_2,Y_2]$ are closed intervals in $B_n(q)$. Since the labeling for $B_n(q)$ is an EL-labeling, there is a unique increasing maximal chain $c_1$ in $[X_1,Y_1]$ that lexicographically precedes all other chains in the same interval. There is also a unique increasing maximal chain~$c_2$ in $[X_2,Y_2]$. Then the chain in $[X,Y]$ formed by pairing elements of $c_1$ and $c_2$ of the same rank must be the unique increasing maximal chain in $[X,Y]$. Any other chain would have non-increasing labels in $[X_1,Y_1]$ or $[X_2,Y_2]$, hence is non-increasing in~$[X,Y]$. This unique increasing maximal chain of $[X,Y]$ must also satisfy part $(2)$ of Definition~\ref{EL} because $c_1$ and $c_2$ both satisfy this condition. 
\end{proof}

The following theorem of Bj\"orner and Wachs connects the permutations in $\mathcal S_n$ with the maximal chains of the Segre product poset $B_n(q)\circ B_n(q)$. Let $\hat{P}$ be the bounded extension of $P$. That is, $\hat{P}=P\cup\{\hat{0},\hat{1}\}$ and $\hat{0}$ and $\hat{1}$ are attached even if $P$ already has a bottom or a top element. 

\begin{thm} \label{EL_homology} \emph{(Bj\"orner and Wachs~\cite{BW4}*{Theorem~4.1}, see also Wachs~\cite{Wachs_notes}*{Theorem~3.2.4})}. Suppose $P$ is a poset for which $\hat{P}$ admits an EL-labeling. Then the order complex of $P$ has the homotopy type of a wedge of spheres, where the number of $i$-spheres is the number of decreasing maximal $(i+2)$-chains of $\hat{P}$. The decreasing maximal $(i+2)$-chains, with $\hat{0}$ and $\hat{1}$ removed, form a basis for the cohomology~$\widetilde H^i(P;\mathbb Z)$.
\end{thm}

Since $P$ has the homotopy type of a wedge of spheres, $\widetilde H^i(P;\mathbb Z)\cong \widetilde H_i(P;\mathbb Z)$ (Wachs~\cite{Wachs_notes}*{Theorem 1.5.1}). We will use $\widetilde H_i(P;\mathbb Z)$, the reduced homology of the order complex~$\Delta (P)$, instead of the cohomology group in this paper.

\begin{exm}
Figure~\ref{B_2(2)} is an EL-labeling of the Segre product poset $B_2(2)\circ B_2(2)$. The left-most chain is increasing with label $(12,12)$. The decreasing (i.e. non-increasing) chains have labels $(12,21)$, $(21,12)$, or $(21,21)$. We use $P_n(q)$ to denote the proper part of the Segre product poset, i.e. $P_n(q)\coloneqq B_n(q)\circ B_n(q)\setminus\{\hat{0},\hat{1}\}$. Then the decreasing chains of $B_2(2)\circ B_2(2)$ with the top and bottom elements removed form a basis of~$\widetilde H^0(P_2(2);\mathbb Z)$.

\begin{figure}\
\centering
\begin{tikzpicture}
  \node (max) at (0,4) {$(\mathbb F_2^2,\mathbb F_2^2)$};
  \node (a) at (-4,2) {$\begin{pmatrix}
													\spn\{\langle 1,0\rangle\},\\
													\spn\{\langle 1,0\rangle\}																							
											 \end{pmatrix}$};
  \node (b) at (-1,2) {$\begin{pmatrix}
													\spn\{\langle 1,0\rangle\},\\
													\spn\{\langle 0,1\rangle\}											
											 \end{pmatrix}$};
	\node (c) at (1.5,2) {\dots\dots};  
	\node (e) at (4,2) {$\begin{pmatrix}
													\spn\{\langle 1,1\rangle\},\\
													\spn\{\langle 1,1\rangle\}											
											 \end{pmatrix}$};
  \node (f) at (-2.1,0.7) {\textcolor{red}{$(1,1)$}};
  \node (g) at (-0.07,1) {\textcolor{red}{$(1,2)$}};
  \node (i) at (2.1,0.7) {\textcolor{red}{$(2,2)$}};
	\node (j) at (-2.1,3.3) {\textcolor{red}{$(2,2)$}};
  \node (k) at (-0.07,3) {\textcolor{red}{$(2,1)$}};
	\node (m) at (2.1,3.3) {\textcolor{red}{$(1,1)$}};
  \node (min) at (0,0) {$(\varnothing,\varnothing)$};
  \draw (min) -- (a) -- (max) -- (b) -- (min)-- (e) -- (max);
\end{tikzpicture}
\caption{An EL-labeling of $B_2(2)\circ B_2(2)$}
\label{B_2(2)}
\end{figure}

\end{exm}



\begin{prop} \label{W_n(q)_meaning} 
Let $W_n(q)=\sum_{(\sigma,\tau)\in\mathcal D_n}{q^{(\inv(\sigma)+\inv(\tau))}},$ where $\mathcal D_n$ denotes the set of pairs of permutations $(\sigma,\tau)\in \mathcal S_n\times\mathcal S_n$ with no common ascent. Then $W_n(q)$ equals the total number of decreasing maximal chains of $P_n(q)\coloneqq B_n(q)\circ B_n(q)\setminus\{\hat{0},\hat{1}\}$ with respect to the labeling described above.  
\end{prop}

\begin{proof}
An edge label $(i,j)\in [n]\times [n] \leq (k,l)$ if and only if $i\leq k$ and $j\leq l$. By the definition of our labeling for $B_n(q)\circ B_n(q)$, there cannot be repeat edge labels along any one chain. So a chain label is decreasing as long as the two components of any two consecutive edge labels do not increase at the same time. Each maximal chain labeling of $B_n(q)\circ B_n(q)$ corresponds to a pair of permutations of $\mathcal S_n$. Then labels of decreasing maximal chains are all pairs of permutations with no common ascent. Given a pair of permutations $(\sigma, \tau)$, the number of maximal chains assigned label $(\sigma, \tau)$ is $q^{\inv(\sigma)}\cdot q^{\inv(\tau)}=q^{(\inv(\sigma)+\inv(\tau))}$ by Lemma~\ref{Chains same perm}. Then the total number of decreasing maximal chains of $P_n(q)$ is 
$$W_n(q)=\sum_{(\sigma,\tau)\in\mathcal D_n}{q^{(\inv(\sigma)+\inv(\tau))}}.$$
\end{proof}

\begin{rem}
The Segre product poset $B_n(q)\circ B_n(q)$ is the $q$-analogue of the Segre product poset $B_n\circ B_n$, agreeing with the formal definition of a $q$-analogue in R.~Simion's paper~\cite{Simion}. She showed that the $q$-analogue of an EL-shellable poset is also EL-shellable. The EL-labeling of $B_n(q)\circ B_n(q)$ we use in this paper provides intuition and a combinatorial interpretation for $W_n(q)$.
\end{rem} 

Bj\"orner and Welker proved that if two pure posets are Cohen-Macaulay, then their Segre product is also Cohen-Macaulay (see~\cite{Segre_rees}*{Theorem 1}). This result in particular proves that the poset $B_n(q)\circ B_n(q)$ is Cohen-Macaulay because $B_n(q)$ is. Corollary~\ref{Segre_EL} says that $B_n(q)\circ B_n(q)$ is shellable, which is a stronger property than the Cohen-Macaulayness. Later in Section~\ref{section_sym_analogue} we will use the fact that the Segre product poset $B_n\circ B_n$ is Cohen-Macaulay. 

\begin{proof}[Proof of Theorem~\ref{Q-CSV}] The poset $P_n(q)=B_n(q)\circ B_n(q)\setminus\{\hat{0},\hat{1}\}$ is pure. By Theorem~\ref{EL_homology}, $P_n(q)$ has the homotopy type of a wedge of $(n-2)$-spheres, and its decreasing maximal $(n-2)$-chains form a basis of the reduced $(n-2)$-nd cohomology. Since $P_n(q)$ is graded and EL-shellable, all reduced homology groups other than the top one vanish (Bj\"orner~\cite{B_Shell_CM}). In Proposition~\ref{W_n(q)_meaning}, we defined $W_n(q)$ to be the total number of decreasing maximal chains of $P_n(q)$. Then using the Euler-Poincar\'e formula~\cite{Wachs_notes}*{Theorem~1.2.8} and Philip Hall's theorem (Stanley~\cite{ec1}*{Proposition 3.8.6}) we get 
\begin{equation} \label{Euler}
\mu_{\widehat{P_n}(q)}(\hat{0},\hat{1})=(-1)^nW_n(q)=\widetilde{\chi}(\Delta(P_n(q))),
\end{equation}
where $\widehat{P_n}(q)=B_n(q)\circ B_n(q)$ denotes $P_n(q)$ with $\hat{0}$ and $\hat{1}$ adjoined.

On the other hand, by the definition of the M\"obius function, $$\mu(\hat{0},\hat{1})=-\sum_{\hat{0}\leq x<\hat{1}}{\mu(\hat{0}, x)}.$$ Each $x$ in $P_n(q)$ is the product of two $k$-dimensional subspaces $X_1, X_2$ of $\mathbb{F}_q^n$, for some $k$ with $0\leq k<n$. The intervals $[\hat{0},X_1]$ and $[\hat{0}, X_2]$ are isomorphic to the poset~$B_k(q)$, hence $\mu(\hat{0}, x)$ is just $\mu_{\widehat{P_k}(q)}(\hat{0}, \hat{1})$, where $P_k(q)= B_k(q)\circ B_k(q)\setminus \{\hat{0},\hat{1}\}$. The number of $k$-dimensional subspaces of $\mathbb{F}_q^n$ is $\qbinom{n}{k}_q$ (Stanley~\cite{ec1}*{Proposition 1.7.2}). So the number of distinct $x=(X_1,X_2)$ where $X_1$ and $X_2$ are $k$-dimensional subspaces is $\qbinom{n}{k}_q^2$. Therefore we have 
$$\mu_{\widehat{P_n}(q)}(\hat{0},\hat{1})=-\sum_{i=0}^{n-1}{\qbinom{n}{i}_q^2\mu_{\widehat{P_i}(q)}(\hat{0},\hat{1})}=-\sum_{i=0}^{n-1}{\qbinom{n}{i}_q^2 (-1)^iW_i(q)}.$$ 
Consequently, $$\sum_{i=0}^{n}{\qbinom{n}{i}_q^2 (-1)^i W_i(q)}=0.$$
By Proposition~\ref{W_n(q)_meaning}, $W_i(q)=\sum_{(\sigma,\tau)\in\mathcal D_i}{q^{(\inv(\sigma)+\inv(\tau))}}$ is the number of decreasing maximal chains of $P_i(q)$, where, as above, $\mathcal D_i$ denotes the set of pairs of permutations $(\sigma,\tau)\in \mathcal S_i\times\mathcal S_i$ with no common ascent.
\end{proof}

\begin{cor} \label{W_n(q)_is_Euler}
The Euler characteristic of the Segre product of the subspace lattice $B_n(q)\circ B_n(q)$ is $(-1)^nW_n(q)$. 
\end{cor}
\begin{proof}
See equation \eqref{Euler} in the proof of Theorem~\ref{Q-CSV}.
\end{proof} 





\section{The product Frobenius characteristic map} \label{section_xch_map}

The Frobenius characteristic map is often used to study representations of the symmetric group.
Here we will define a product Frobenius characteristic map to help understand representations of $\mathcal S_n\times \mathcal S_n$. Therefore, let us consider two sets of variables $x=(x_1, x_2,...)$ and $y=(y_1, y_2,...)$. Following Sagan's notations~\cite{Sagan}, $R^n$ denotes the space of class functions on $\mathcal S_n$ and $R=\oplus_nR^n$. We will use $R^{m,n}$ to denote the space of class functions on $\mathcal S_m\times \mathcal S_n$ and let $R_{2d}=\oplus_{m,n}R^{m,n}$. Let $\Lambda^n$ be the space of homogeneous degree $n$ symmetric functions. Then $\Lambda(x)=\oplus_n\Lambda^n(x)$ and $\Lambda(y)=\oplus_n\Lambda^n(y)$ denote the rings of symmetric functions in variables $(x_1, x_2,...)$ and $(y_1,y_2,...)$ respectively. Given $\mu\vdash n$ with $\mu=(1^{m_1}2^{m_2}\dots)$, we write $z_{\mu}=\prod_{i=1}^{i=n}{i^{m_i}m_i!}$.

Let us recall the definition of the usual characteristic map.
\begin{dfn} \label{ch_def}
The \emph{(Frobenius) characteristic map} $\textrm{ch}^n : R^n\to\Lambda^n$ is defined by
$$\textrm{ch}^n(\chi)=\sum_{\mu\vdash n}z_{\mu}^{-1}\chi_{\mu}p_{\mu},$$
where $\chi_{\mu}$ is the value of $\chi$ on the class $\mu$ and $p_{\mu}$ is the power sum symmetric function. Define $\textrm{ch}\coloneqq \oplus_n\textrm{ch}^n$.
\end{dfn}

Now we define a product characteristic map.
\begin{dfn} 
Let $\chi$ be a class function on $\mathcal S_m\times\mathcal S_n$. The \emph{product Frobenius characteristic map} $\textrm{ch}: R_{2d}\to\Lambda(x) \otimes\Lambda(y)$ is defined as: 
\begin{equation} \label{prod_ch_formula}
\textrm{ch}(\chi)=\sum_{(\mu,\lambda)\vdash (m,n)}z_{\mu}^{-1}z_{\lambda}^{-1}\chi_{(\mu,\lambda)}p_{\mu}(x)p_{\lambda}(y),
\end{equation}
where $\chi_{(\mu,\lambda)}$ is the value of $\chi$ on the class $(\mu,\lambda)$ and $p_{\mu}$, $p_{\lambda}$ are power sum symmetric functions. The class $(\mu,\lambda)$ is indexed by a partition $\mu\vdash m$ and a partition $\lambda\vdash n$ that tell us the cycle types of elements of $\mathcal S_m$ and $\mathcal S_n$ respectively.
\end{dfn}

\begin{prop} \label{prod_ch_tensor} For a character $f\otimes g$ of $\mathcal S_m\times\mathcal S_n$, where $f$ is a character of~$\mathcal S_m$ and $g$ a character of $\mathcal S_n$, the product Frobenius characteristic $\mathrm{ch}(f\otimes g)$ equals $\mathrm{ch}(f)(x)\mathrm{ch}(g)(y)$. 
\end{prop}

\begin{proof}
Equation \eqref{prod_ch_formula} gives us
$$\textrm{ch}(f\otimes g)=\sum_{(\mu,\lambda)\vdash (m,n)}z_{\mu}^{-1}z_{\lambda}^{-1}(f\otimes g)_{(\mu,\lambda)}p_{\mu}(x)p_{\lambda}(y).$$
For a conjugacy class $(\mu,\lambda)\vdash (m,n)$, let $\sigma\in\mathcal S_m$ have cycle type $\mu$ and $\tau\in\mathcal S_n$ have cycle type $\lambda$. The character value
$$(f\otimes g)_{(\mu,\lambda)}=(f\otimes g)(\sigma,\tau)=f(\sigma)g(\tau)=f_{\mu}g_{\lambda},$$
where the second equality is by~\cite{Sagan}*{Theorem 1.11.2}. Then
\begin{align*}
\textrm{ch}(f\otimes g)&=\sum_{(\mu,\lambda)\vdash (m,n)}z_{\mu}^{-1}z_{\lambda}^{-1}f_{\mu}g_{\lambda}p_{\mu}(x)p_{\lambda}(y)\\
&=\sum_{\mu\vdash m}z_{\mu}^{-1}f_{\mu}p_{\mu}(x)\sum_{\lambda\vdash n}z_{\lambda}^{-1}g_{\lambda}p_{\lambda}(y)\\
&=\textrm{ch}(f)(x)\textrm{ch}(g)(y).
\end{align*}
\end{proof}

Because the product Frobenius characteristic map is an extension of the usual (Frobenius) characteristic map, we keep the notation $\textrm{ch}$ for product Frobenius characteristic map even though $\textrm{ch}$ was previously defined to be $\oplus_n\textrm{ch}^n$ in various literature (Sagan~\cite{Sagan}, Stanley~\cite{ec2}).  The meaning of $\textrm{ch}$ will be clear in the given context.

Recall that the \emph{induction product} $f\circ g$ is the induction of $f\otimes g$ from $\mathcal S_m\times\mathcal S_n$ to~$\mathcal S_{m+n}$. A fundamental property of the usual characteristic map is the following:

\begin{prop} \label{regular_chch} \emph{(Stanley~\cite{ec2}*{Proposition 7.18.2})} The Frobenius characteristic map $\mathrm{ch}: R\to \Lambda$ is a bijective ring homomorphism, i.e., \emph{ch} is one-to-one and onto, and satisfies $$\mathrm{ch}(f\circ g)=\mathrm{ch}(f)\mathrm{ch}(g).$$
\end{prop}

\begin{rem}\label{tensor_vs_induction}
The product Frobenius characteristic on the tensor product of characters, $\textrm{ch}(f\otimes g)=\textrm{ch}(f)(x)\textrm{ch}(g)(y)$, is a symmetric function in $\Lambda^m(x)\otimes\Lambda^n(y)$, while the usual Frobenius characteristic on the induction product of characters, $\textrm{ch}(f\circ g)=\textrm{ch}(f)(x)\textrm{ch}(g)(x)$, is a symmetric function in $\Lambda^{m+n}$. 
\end{rem}

We would like the product Frobenius characteristic map to be a homomorphism as well. Given an $\mathcal S_k\times\mathcal S_l$-module $V$ with character $\psi$ and an $\mathcal S_m\times\mathcal S_n$-module $W$ with character $\phi$, $\psi\otimes\phi$ is the character of $V\otimes W$, which is a representation of $(\mathcal S_k\times\mathcal S_l)\times(\mathcal S_m\times\mathcal S_n)$. We want to produce a character of $\mathcal S_{k+m}\times\mathcal S_{l+n}$.

\begin{dfn}
For $\psi$ and $\phi$ as given above, we define the \emph{induction product} $\psi\circ\phi$ to be $\psi\otimes\phi\uparrow_{(\mathcal S_k\times\mathcal S_l)\times(\mathcal S_m\times\mathcal S_n)}^{\mathcal S_{k+m}\times\mathcal S_{l+n}}$. The induction product on characters extends to all class functions on $R_{2d}$ by (bi)linearity.
\end{dfn}


\begin{prop} \label{prod_ch}
Let $\psi$ be a class function on $\mathcal S_k\times\mathcal S_l$, and $\phi$ a class function on $\mathcal S_m\times\mathcal S_n$. The product Frobenius characteristic map $\mathrm{ch}: R_{2d}\to \Lambda(x)\otimes\Lambda(y)$ is a bijective ring homomorphism, i.e., \emph{ch} is one-to-one and onto, and satisfies $$\mathrm{ch}(\psi\circ\phi)=\mathrm{ch}(\psi)\mathrm{ch}(\phi).$$
\end{prop}


Before proving this proposition, we need to establish the following lemma:

\begin{lem} \label{tensor_ind}
Given two groups $A$ and $B$, and their subgroups $F<A$ and $G<B$, if~$f$ is the character of a representation of $F$ and $g$ is the character of a representation of~$G$, then 
$$f\otimes g\uparrow_{F\times G}^{A\times B}=f\uparrow_{F}^{A}\otimes g\uparrow_{G}^{B}.$$
\end{lem}

\begin{proof}
Suppose $F<A$ has coset representatives $\{s_1,s_2,...,s_q\}$, and $G<B$ has coset representatives $\{t_1,t_2,...,t_r\}$. Then $\{(s_i, t_j):i\in [q], j\in [r]\}$ is a set of coset representatives for $F\times G< A\times B$. For $(\sigma,\tau)\in A\times B$,
\begin{align*}
f\otimes g\uparrow_{F\times G}^{A\times B}((\sigma,\tau)) &=\sum_{i,j}f\otimes g\big((s_i^{-1},t_j^{-1})(\sigma,\tau)(s_i,t_j)\big)\\
& = \sum_i{f(s_i^{-1}\sigma s_i)}\sum_j{g(t_j^{-1}\tau t_j)}\\
&=f\uparrow_{F}^{A}(\sigma) g\uparrow_{G}^{B}(\tau)\\
&=f\uparrow_{F}^{A}\otimes g\uparrow_{G}^{B}((\sigma,\tau)).
\end{align*}
For the second and fourth equalities, see~\cite{Sagan}*{Theorem 1.11.2}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prod_ch}] 
The bijectiveness of the product Frobenius characteristic map follows from the definition of $\textrm{ch}$ and the fact that the power sums $p_\mu(x)p_\lambda(y)$ form a $\mathbb{Q}$-basis for $\Lambda(x)\otimes\Lambda(y)$. Next we will show that the map is a homomorphism. Suppose $\psi=\sum_{i,j}a_{ij}\psi_k^{(i)}\otimes \psi_l^{(j)}$ such that $\psi_k^{(i)}$'s and $\psi_l^{(j)}$'s are irreducible characters of representations of $\mathcal S_k$ and $\mathcal S_l$ respectively. Similarly, $\phi =\sum_{u,v}b_{uv}\phi_m^{(u)}\otimes\phi_n^{(v)}$. For any $\sigma_k\in\mathcal S_k$, $\sigma_l\in\mathcal S_l$, $\tau_m\in\mathcal S_m$, and $\tau_n\in\mathcal S_n$, we have
\begin{align*}
\psi\otimes\phi\big((\sigma_k,\sigma_l),(\tau_m,\tau_n)\big) & = \big(\sum_{i,j}a_{ij}\psi_k^{(i)}(\sigma_k)\psi_l^{(j)}(\sigma_l)\big)\big(\sum_{u,v}b_{uv}\phi_m^{(u)}(\tau_m)\phi_n^{(v)}(\tau_n)\big)\\
&=\sum_{i,j,u,v}a_{ij}b_{uv}\psi_k^{(i)}(\sigma_k)\phi_m^{(u)}(\tau_m)\psi_l^{(j)}(\sigma_l)\phi_n^{(v)}(\tau_n)\\
&=\sum_{i,j,u,v}a_{ij}b_{uv}(\psi_k^{(i)}\otimes\phi_m^{(u)})\otimes(\psi_l^{(j)}\otimes\phi_n^{(v)})(\sigma_k,\tau_m,\sigma_l,\tau_n).
\end{align*}
Thus, $\psi\otimes\phi=\sum_{i,j,u,v}a_{ij}b_{uv}(\psi_k^{(i)}\otimes\phi_m^{(u)})\otimes(\psi_l^{(j)}\otimes\phi_n^{(v)})$. So, 
\begin{align*}
\psi\circ\phi & =\psi\otimes\phi\uparrow_{(\mathcal S_k\times\mathcal S_l)\times(\mathcal S_m\times\mathcal S_n)}^{\mathcal S_{k+m}\times\mathcal S_{l+n}}\\
& =\sum_{i,j,u,v}a_{ij}b_{uv}(\psi_k^{(i)}\otimes\phi_m^{(u)})\otimes(\psi_l^{(j)}\otimes\phi_n^{(v)})\uparrow_{\mathcal S_k\times\mathcal S_m\times\mathcal S_l\times\mathcal S_n}^{\mathcal S_{k+m}\times\mathcal S_{l+n}}\\
& =\sum_{i,j,u,v}a_{ij}b_{uv}(\psi_k^{(i)}\otimes\phi_m^{(u)})\uparrow_{\mathcal S_k\times\mathcal S_m}^{\mathcal S_{k+m}}\otimes (\psi_l^{(j)}\otimes\phi_n^{(v)})\uparrow_{\mathcal S_l\times\mathcal S_n}^{\mathcal S_{l+n}}\\ 
& = \sum_{i,j,u,v}a_{ij}b_{uv}(\psi_k^{(i)}\circ\phi_m^{(u)})\otimes (\psi_l^{(j)}\circ\phi_n^{(v)})
\end{align*}
by Lemma~\ref{tensor_ind}. Now take the product Frobenius characteristic of both sides of the above equation. For clarity, we keep track of variables $x$ and $y$. By Proposition~\ref{prod_ch_tensor} and then Proposition~\ref{regular_chch} we get
\begin{align*}
\textrm{ch}(\psi\circ\phi)(x,y)&=\sum_{i,j,u,v}a_{ij}b_{uv}\textrm{ch}(\psi_k^{(i)}\circ\phi_m^{(u)})(x)\textrm{ch}(\psi_l^{(j)}\circ\phi_n^{(v)})(y)\\
&=\sum_{i,j,u,v}a_{ij}b_{uv}\textrm{ch}(\psi_k^{(i)})(x)\textrm{ch}(\phi_m^{(u)})(x)\textrm{ch}(\psi_l^{(j)})(y)\textrm{ch}(\phi_n^{(v)})(y)\\
&=\sum_{i,j}a_{ij}\textrm{ch}(\psi_k^{(i)})(x)\textrm{ch}(\psi_l^{(j)})(y)\sum_{u,v}b_{uv}\textrm{ch}(\phi_m^{(u)})(x)\textrm{ch}(\phi_n^{(v)})(y)\\
&=\textrm{ch}(\psi)(x,y)\textrm{ch}(\phi)(x,y).
\end{align*} 
\end{proof}




\section{A symmetric function analogue} \label{section_sym_analogue}

Using the product Frobenius characteristic map, we derive an equation that is analogous to a well-known symmetric function identity (see Stanley~\cite{ec2}*{equation (7.13)}): for $n\geq 1$, 
$$\sum_{i=0}^n(-1)^ie_ih_{n-i}=0.$$ 

The above identity contains the complete homogeneous symmetric function $h_{n-i}$ and the elementary symmetric function $e_i$, which is the Frobenius characteristic of the representation of $\mathcal S_i$ on the top homology of the subset lattice $B_i$. Our analogue, equation~\eqref{hhch}, involves $h_{n-i}(x)h_{n-i}(y)$ and the representation of $\mathcal S_n\times\mathcal S_n$ on the top homology of the Segre product poset $B_n\circ B_n$. The product $\mathcal S_n\times\mathcal S_n$ acts on $B_n\circ B_n$ using the usual action of $\mathcal S_n$ on $B_n$ in each component separately. For instance, given a pair of permutations $(123, 213)\in \mathcal S_3\times\mathcal S_3$ written in one-line notation and an element $(\{1,2\},\{2,3\})\in B_3\circ B_3$, the first permutation $123$ fixes the subset $\{1,2\}$ and the second permutation $213$ takes $\{2,3\}$ to $\{1,3\}$ by permuting the numbers in the subset. In the proof of our analogue, we use the Whitney homology technique, which was introduced by Sundaram~\cite{Sheila2} for pure posets and then generalized by Wachs~\cite{Wachs2} for semipure posets.

Let $Q$ be a poset with a bottom element $\hat{0}$ and $G$ an automorphism group of~$Q$. Suppose $Q$ is a Cohen-Macaulay $G$-poset, for each integer $r$, the $r$-th \emph{Whitney homology} of $Q$ is defined as $$WH_r(Q)=\bigoplus_{x\in Q_r}\widetilde{H}_{r-2}(\hat{0},x),$$ where $Q_r\coloneqq \{x\in Q\mid  \rank(x)=r\}$.

For the subset lattice $B_n$, let $P_n$ be the proper part of the Segre product poset $B_n\circ B_n$. The action of $\mathcal{S}_n \times \mathcal{S}_n$ on $P_n$ induces a representation on the reduced top homology of $P_n$.

\begin{thm} \label{Stanley q-form}
Let $\mathrm{ch}(\widetilde H_{n-2}(P_n))$ be the product Frobenius characteristic of this representation. Then 
\begin{equation} \label{hhch}
{\sum_{i=0}^{n}}{(-1)^ih_{n-i}(x)h_{n-i}(y)\mathrm{ch}(\widetilde H_{i-2}(P_i))}=0.
\end{equation}
\end{thm}

\begin{proof}
Let $Q$ be $P_{n}\cup{\hat{0}}$, which is Cohen-Macaulay. We consider the Whitney homology of $Q$. From the work of Sundaram on Whitney homology (Sundaram \cites{Sheila1, Sheila2}, Wachs~\cite{Wachs_notes}*{Theorem 4.4.1}), we know that 

$$\widetilde H_{n-2}(P_{n})\cong_{\mathcal S_n\times\mathcal S_n}{\displaystyle \bigoplus_{r=0}^{n-1}}(-1)^{n-1+r}\mbox{WH}_r(Q).$$ 

The action of $\mathcal S_{n} \times \mathcal S_{n}$ on $Q$ induces a representation of $\mathcal S_{n} \times \mathcal S_{n}$ on the reduced top homology of $Q$ and its Whitney homology groups. An interval $[\hat{0},x]$ is taken to~$[\hat{0},(\sigma,\tau)x]$ for $(\sigma,\tau)\in \mathcal S_n\times \mathcal S_n$. Both $\widetilde H_{n-2}(P_{n})$ and $\mbox{WH}_r(Q)$ are $\mathcal S_n\times\mathcal S_n$-modules. Let $x$ be a rank $r$ element of $Q$. The stabilizer of $x$ is then the Young subgroup $(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})$. Viewing the Whitney homology groups as $\mathcal S_n\times\mathcal S_n$-modules,   

$$\mbox{WH}_r(Q)={\displaystyle \bigoplus_{x\in Q_r/(\mathcal S_{n}\times \mathcal S_{n})}}\widetilde H_{r-2}(\hat{0},x)\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}},$$ 
where $Q_r$ is the set of rank $r$ elements in $Q$ and $Q_r/(\mathcal S_{n} \times \mathcal S_{n})$ is a set of orbit representatives in $Q_r$ (see Wachs~\cite[Lecture $4.4$]{Wachs_notes}). The action of $\mathcal S_{n} \times \mathcal S_{n}$ on $Q_r$ is transitive. So the contribution of the $r$-th Whitney homology to $\widetilde H_{n-2}(P_{n})$ is the induced representation $\widetilde H_{r-2}(\hat{0},x)\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (S_{r}\times S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}}$ for any $x$ in $Q_r$. The open interval $(\hat{0}, x)$ is isomorphic to the poset $P_r$. We then have
$$\mbox{WH}_r(Q)=\widetilde H_{r-2}(P_{r})\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}},$$ 
and
$$\widetilde H_{n-2}(P_{n})\cong_{\mathcal S_{n}\times \mathcal S_{n}}{\displaystyle \bigoplus_{r=0}^{n-1}}(-1)^{n-1+r}\widetilde H_{r-2}(P_{r})\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}}.$$ 

Taking the product Frobenius characteristic of both sides of the above equation, we get
\begin{equation}\label{ch_Whitney} \textrm{ch}(\widetilde H_{n-2}(P_{n}))={\displaystyle \sum_{r=0}^{n-1}}(-1)^{n-1+r}ch\big(\widetilde H_{r-2}(P_{r})\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}}\big).\end{equation} 

Now let $\psi_r$ be the character of the $(\mathcal S_r\times \mathcal S_r)$-module $\widetilde H_{r-2}(P_{r})$. Write $1_{\mathcal S_{n-r}\times \mathcal S_{n-r}}$ for the character of the trivial representation of ${S_{n-r}\times S_{n-r}}$. When viewing $\widetilde H_{r-2}(P_{r})$ as a $(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})$-module, its character equals $\psi_r\otimes 1_{\mathcal S_{n-r}\times \mathcal S_{n-r}}$ (Sagan~\cite{Sagan}*{Theorem 1.11.2}). 
Then
\begin{align*}
\widetilde H_{r-2}(P_{r})\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}} & =\psi_r\otimes 1_{\mathcal S_{n-r}\times \mathcal S_{n-r}}\uparrow_{(\mathcal S_{r}\times \mathcal S_{n-r})\times (\mathcal S_{r}\times \mathcal S_{n-r})}^{\mathcal S_{n} \times \mathcal S_{n}}\\
 & =\psi_r\circ 1_{\mathcal S_{n-r}\times \mathcal S_{n-r}}.
\end{align*}

It follows from Proposition~\ref{prod_ch} that the product Frobenius characteristic $$\textrm{ch}(\psi_r\circ 1_{\mathcal S_{n-r}\times \mathcal S_{n-r}})=\textrm{ch}(\psi_r)\textrm{ch}(1_{\mathcal S_{n-r}\times \mathcal S_{n-r}}).$$ 

Thus, equation (\ref{ch_Whitney}) becomes 
\begin{equation}\label{ch_H_decomp}\begin{split}
\textrm{ch}(\widetilde H_{n-2}(P_{n})) & ={\displaystyle \sum_{r=0}^{n-1}}{(-1)^{n-1+r}\textrm{ch}(\widetilde H_{r-2}(P_r))\textrm{ch}(1_{\mathcal S_{n-r}\times \mathcal S_{n-r}})}\\
& = {\displaystyle \sum_{r=0}^{n-1}}{(-1)^{n-1+r}\textrm{ch}(\widetilde H_{r-2}(P_r))\textrm{ch}(1_{\mathcal S_{n-r}})(x)\textrm{ch}(1_{\mathcal S_{n-r}})(y)}.
\end{split}\end{equation}

It is known that the Frobenius characteristic of the trivial representation of $\mathcal{S}_n$ is~$h_n$ (Stanley~\cite{ec2}). Multiplying both sides of equation (\ref{ch_H_decomp}) by $(-1)^{n-1}$, we get
$$(-1)^{n-1}\textrm{ch}(\widetilde H_{n-2}(P_{n})) ={\displaystyle \sum_{r=0}^{n-1}}{(-1)^{r}\textrm{ch}(\widetilde H_{r-2}(P_r))h_{n-r}(x)h_{n-r}(y)}.$$

Finally, we conclude that $$\sum_{i=0}^{n}{(-1)^ih_{n-i}(x)h_{n-i}(y)\textrm{ch}(\widetilde H_{i-2}(P_i))}=0.$$
\end{proof}

Let $ps:\Lambda \to \mathbb Q [q]$ be the stable principal specialization, that is, for a symmetric function $f(x_1,x_2,x_3,\dots)$, $\mathrm{ps}(f)$ is defined to be $f(1,q,q^2,...)$. A summary of the specializations of different bases for the symmetric functions can be found in Stanley~\cite{ec2}*{Proposition $7.8.3$}. Consider a symmetric function $f$ in two sets of variables $(x_1,x_2,...)$ and $(y_1,y_2,...)$. We take the stable principal specialization of $f$ in each set of variables, i.e. substitute $(1,q,q^2,...)$ for both $(x_1,x_2,...)$ and $(y_1,y_2,...)$. The product Frobenius characteristic of the $\mathcal S_n\times \mathcal S_n$-modules $\widetilde H_{n-2}(P_n)$ is a symmetric function in two sets of variables. Then it is natural to ask what we can say about its specialization. 

Recall that $P_n$ is the proper part of the Segre product of the subset lattice $B_n$ with itself. The product Frobenius characteristic of the $\mathcal S_n\times \mathcal S_n$-module $\widetilde H_{n-2}(P_n)$ has an innate connection with the Euler characteristic of $B_n(q)\circ B_n(q)$. From Corollary \ref {W_n(q)_is_Euler}, $W_n(q)$ is the signless Euler characteristic of $B_n(q)\circ B_n(q)$. The following theorem gives us a connection between the stable principal specialization of $\textrm{ch}(\widetilde H_{n-2}(P_n))$ and the Euler characteristic $W_n(q)$.
\begin{thm}\label{sp_ch_pn}
Let $W_n(q)$ be the signless Euler characteristic of $B_n(q)\circ B_n(q)$. For a symmetric function $f$ in two sets of variables $x=(x_1, x_2, \dots)$ and $y=(y_1,y_2,\dots)$, the stable principal specialization $\mathrm{ps}(f)$ specializes both $x_i$ and $y_i$ to $q^{i-1}$. Then $$\mathrm{ps}(\mathrm{ch}(\widetilde{H}_{n-2}(P_n)))=\frac{W_n(q)}{\displaystyle{\prod_{i=1}^{n}{(1-q^i)^2}}},$$ where $\mathrm{ch}(V)$ is the product Frobenius characteristic of $V$.
\end{thm}

\begin{proof}
We will use induction. The base cases $n=2$ and $n=3$ can be verified by hand. We can compute that
$$\mathrm{ps}(\textrm{ch}(\widetilde H_0(P_2)))=\frac{q^2+2q}{(1-q)^2(1-q^2)^2}=\frac{W_2(q)}{(1-q)^2(1-q^2)^2},$$ and 
$$\mathrm{ps}(\textrm{ch}(\widetilde H_1(P_3)))=\frac{q^6+4q^5+6q^4+6q^3+2q^2}{(1-q)^2(1-q^2)^2(1-q^3)^2}=\frac{W_3(q)}{(1-q)^2(1-q^2)^2(1-q^3)^2}.$$ Assume that the statement is true for $P_i, i=1,...,n-1$. Now let us consider the reduced top homology of $P_{n}$. Equation (\ref{hhch}) gives us a way to express $\textrm{ch}(\widetilde H_{n-2}(P_n))$ in terms of the product Frobenius characteristic of smaller posets. We get
\begin{equation} \label{ch}
\textrm{ch}(\widetilde H_{n-2}(P_n))= \sum_{i=0}^{n-1}(-1)^{n-1+i}h_{n-i}(x)h_{n-i}(y)\textrm{ch}(\widetilde H_{i-2}(P_i)).
\end{equation}

Then we take the stable principal specialization of both sides of equation (\ref{ch}). We know from Stanley~\cite{ec2} that $\mathrm{ps}(h_{n})=\prod_{i=1}^{n}{\frac{1}{1-q^i}}$. It follows from our induction hypothesis that
\begin{equation*}
\begin{split}
\mathrm{ps}(\textrm{ch}(\widetilde H_{n-2}(P_n))) & = \sum_{i=0}^{n-1}(-1)^{n-1+i}\mathrm{ps}(\textrm{ch}(\widetilde H_{i-2}(P_i)))\prod_{j=1}^{n-i}{\frac{1}{(1-q^j)^2}}\\
& = \sum_{i=0}^{n-1}(-1)^{n-1+i}\frac{W_i(q)}{\prod_{k=1}^{i}{(1-q^k)^2}}\prod_{j=1}^{n-i}{\frac{1}{(1-q^j)^2}}\\
& = \frac{1}{\prod_{k=1}^{n}{(1-q^k)^2}}\cdot \sum_{i=0}^{n-1}{(-1)^{n-1+i}W_i(q)\frac{\prod_{j=i+1}^{n}{(1-q^j)^2}}{\prod_{j=1}^{n-i}{(1-q^j)^2}}}\\
& = \frac{1}{\prod_{k=1}^{n}{(1-q^k)^2}}\cdot \sum_{i=0}^{n-1}{(-1)^{n-1+i}W_i(q)\qbinom{n}{i}_q^2}.
\end{split}
\end{equation*}

Finally, using the identity involving the signless Euler characteristic $W_n(q)$ given in Theorem~\ref{Q-CSV}, we obtain $$\mathrm{ps}(\textrm{ch}(\widetilde H_{n-2}(P_{n})))=\frac{W_{n}(q)}{\prod_{j=1}^{n}{(1-q^j)^2}}.$$
\end{proof}

Theorem~\ref{Stanley q-form} was motivated by our initial findings regarding the $q$-analogue of equation \eqref{CSV_intro}. Once we formulated the specialization of $\textrm{ch}(\widetilde H_{i-2}(P_i))$, the $q$-analogue can be retrieved by taking the stable principal specialization of equation \eqref{hhch}. 




\section{Alternative proof of the result of Carlitz--Scoville--Vaughan} \label{alt_proof_CSV}

Carlitz, Scoville and Vaughan's result, Theorem~\ref{CSV_Othm}, provides a combinatorial explanation for the coefficients $\omega_k$ in the reciprocal Bessel function. They showed that $\omega_k$ is the number of pairs of $k$-permutations with no common ascent. When letting $q=1$ in our $q$-analogue \eqref{W_n(q)_intro}, the subspaces of $\mathbb F_q^n$ become subsets of $\{1,2,...,n\}$. The value $W_n(1)=\sum_{(\sigma, \tau)\in \mathcal D_n} 1^{\inv(\sigma)+\inv(\tau)}$ simply counts the number of pairs of permutations of $[n]$ with no common ascent, i.e. $\omega_n$. The proof of Theorem~\ref{Q-CSV} is then easily adapted into an alternative proof of Carlitz, Scoville and Vaughan's result \eqref{CSV_intro}. Carlitz, Scoville and Vaughan's proof in~\cite{CSV} includes general cases where occurrences of common ascent are allowed. Our proof does not account for those general cases, but it gives a less technical approach by utilizing Bj\"{o}rner and Wachs' work on shellability and poset homology~\cite{BW4}.  

 
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