co.combinatorics - Generating functions for certain statistics on Coxeter groups of type B

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

$$sum_{win S_{n}}t{}^{l(w)}=prod_{i=1}^{n}frac{1-t^{i}}{1-t},$$

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

$$D_{R}(w):={xin Wmid l(wx)<l(w)}$$

is contained in a given subset $Isubseteq S$ of the fundamental
reflections $S$ of the group $W$. There are several examples in
the literature of sums of the form

$$sum_{substack{win W\ D_{R}(w)subseteq I} }t^{f(w)}quadtext{or}sum_{substack{win W\ D_{R}(w)subseteq I} }(-1)^{l(w)}t^{f(w)},$$

where $f:Wrightarrowmathbb{N}$ 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 $win S_{n}^{B}$, we write $w=[a_{1},dots,a_{n}]$
to mean $w(i)=a_{i}$, for $i=1,dots,n$. For $iin[n-1]$, 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 $Isubseteq[n-1]_{0}$,
we write $D_{R}(w)subseteq I$ rather than $D_{R}(w)subseteq{s_{i}mid iin I}$.

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):

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

Here $lfloorcdotrfloor$ denotes the floor function, and we use
the notation $(underline{0}):=1$, $(underline{a}):=1-t^{a}$, for
$ageq1$, 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 $S_{n}^{B}$:

$$tilde{L}(w):=frac{1}{2}|{x,yin[pm n]mid x<y, w(x)>w(y), xnotequiv ypmod{2}}|.$$

The question is now:

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

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

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

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

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