Background

In combinatorics one is sometimes interested in various 'statistics' on a Coxeter group (e.g., functions from the group to the natural numbers), and to find a 'nice' expression for a corresponding generating function. For example, the length function $l$ on a Coxeter group $W$ is an important statistic, and when $W=S_{n}$ is the symmetric group on $n$ letters, a classical result of this type is the identity $$\sum_{w\in S_{n}}t{}^{l(w)}=\prod_{i=1}^{n}\frac{1-t^{i}}{1-t},$$

where $t$ is an indeterminate (cf. Stanley, Enumerative Combinatorics, vol. 1, Coroll. 1.3.10). There are also variations on the problem, where one considers sums over elements $w$ whose right descent set

$$D_{R}(w):=\{x\in W\mid l(wx)<l(w)\}$$

is contained in a given subset $I\subseteq S$ of the fundamental reflections $S$ of the group $W$. There are several examples in the literature of sums of the form $$ \sum_{\substack{w\in W\\ D_{R}(w)\subseteq I} }t^{f(w)}\quad\text{or}\sum_{\substack{w\in W\\ D_{R}(w)\subseteq I} }(-1)^{l(w)}t^{f(w)},$$

where $f:W\rightarrow\mathbb{N}$ is a given statistic on $W$, and it is sometimes possible to express these generating functions in a (non-trivial) simple algebraic way, as in the above example.

Let $[n]$ denote the set $\{1,2,\dots,n\}$, and let $S_{n}^{B}$ be the signed permutation group, that is, the group of all bijections $w$ of the set $[\pm n]=\{\pm1,\pm2,\dots,\pm n\}$, such that $w(-a)=-a$, for all $a$ in the set (cf. Björner \& Brenti: Combinatorics of Coxeter Groups, 8.1). If $w\in S_{n}^{B}$, we write $w=[a_{1},\dots,a_{n}]$ to mean $w(i)=a_{i}$, for $i=1,\dots,n$. For $i\in[n-1]$, the $i$th Coxeter generator of $S_{n}^{B}$ is given by $$ s_{i}:=[1,\dots,i-1,i+1,i,i+2,\dots,n],$$ and we also put $$s_{0}:=[-1,2,\dots,n].$$

We may therefore identify the set of generators $s_{i}$ with the set $[n-1]_{0}:=[n-1]\cup\{0\}$. Hence, for any subset $I\subseteq[n-1]_{0}$, we write $D_{R}(w)\subseteq I$ rather than $D_{R}(w)\subseteq\{s_{i}\mid i\in I\}$.

Questions

In addition to defining a collection of generators, a set $I=\{i_{1},\dots,i_{l}\}\subseteq[n-1]_{0}$, with $i_{1}<i_{2}<\cdots<i_{l}$ also defines the following polynomial (related to Gaussian polynomials):

$$\alpha_{I,n}(t):=\frac{(\underline{n})!}{(\underline{i_{1}})!\prod_{r=1}^{l}\prod_{s=1}^{\lfloor(i_{r+1}-i_{r})/2\rfloor}(\underline{2s})}.$$

Here $\lfloor\cdot\rfloor$ denotes the floor function, and we use the notation $(\underline{0}):=1$, $(\underline{a}):=1-t^{a}$, for $a\geq1$, and $(\underline{a})!:=(\underline{1})(\underline{2})\cdots(\underline{a})$. To get a correct formula, we also put $i_{l+1}:=n$.}

Define the following statistic on $S_{n}^{B}$: $$\tilde{L}(w):=\frac{1}{2}|\{x,y\in[\pm n]\mid x<y,\ w(x)>w(y),\ x\not\equiv y\pmod{2}\}|.$$

The question is now:

Is it true that for any $n$ and $I$ as above, we have

$$\alpha_{I,n}(t)=\sum_{\substack{w\in S_{n}^{B}\\ D_{R}(w)\subseteq I} }(-1)^{l(w)}t^{\tilde{L}(w)}?$$

A less precise but more general question is the following: Given a family of polynomials $p_{I,n}(t)\in\mathbf{Z}[t]$ depending on $I$ and $n$, is there any general (non-trivial) sufficient criterion for the existence of functions $f,g:W\rightarrow\mathbb{N}$ on a finite Coxeter group $W$, such that for all $I$ and $n$, we have

$$p_{I,n}(t)=\sum_{\substack{w\in W\\ D_{R}(w)\subseteq I} }a^{f(w)}t^{g(w)},$$ for some $a\in\mathbf{Z}$?

  • It's a pity to see you have put in so much effort in posing this question and no one seems to know how to answer it... – Max Muller May 16 '10 at 21:52

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