%% An AMS-TeX file for a 19 page document
\input amstex
\magnification=\magstep1
\NoBlackBoxes
\define \integers {\Bbb Z}
\define \reals {\Bbb R}
\define \nonnegs {\Bbb N}
\define \affS {\tilde{S}}
\define \affA {\tilde{A}}
\define \affC {\tilde{C}}
\define \affB {\tilde{B}}
\define \affD {\tilde{D}}
\define \des {\roman {des}}
\define \Des {\roman {Des}}
\define \dex {\roman {dex}}
\define \CC#1#2{\Delta(#1,#2)}
\define \WW#1 { W^{(#1)} }
\define \WWW#1#2 { W^{(#1,#2)} }
\documentstyle{amsppt}
\topmatter
\title The distribution of descents and length in a Coxeter group
\endtitle
\author Victor Reiner
\endauthor
\affil University of Minnesota
\endaffil
\endtopmatter
\centerline{e-mail: {\tt reiner\@math.umn.edu}}
\centerline{Submitted August 19, 1995; accepted November 25, 1995}
\topmatter
\thanks Work supported by
Mathematical Sciences Postdoctoral
Research Fellowship DMS-9206371
\endthanks
\address School of Mathematics, University of Minnesota, Minneapolis, MN 55455
\endaddress
\subjclass 05A15, 33C80
\endsubjclass
\abstract We give a method for computing the $q$-{\it Eulerian distribution}
$$
W(t,q)=\sum_{w \in W} t^{\des(w)} q^{l(w)}
$$
as a rational function in $t$ and $q$,
where $(W,S)$ is an arbitrary Coxeter system, $l(w)$ is the {\it length
function} in $W$, and $\des(w)$ is the number of simple reflections
$s \in S$ for which $l(ws)1
{\smcp the electronic journal of combinatorics 2 (1995),
\#R25\hfill\folio}
\fi}
\document
\subheading{I. Introduction}
Let $(W,S)$ be a Coxeter system (see \cite{Hu} for definitions
and terminology). There are two statistics on elements of the
Coxeter group $W$
$$
\aligned
l(w) &= min\{l:w = s_{i_1} s_{i_2} \cdots s_{i_l} \text{ for some }
s_{i_k} \in S\}\\
\des(w) &= |\{s \in S: l(ws)0\}.
$$
Also denote by $|T|$ the {\it cardinality} $\sum_{s \in S}T(s)$ of
the multiset or function.
\proclaim{Theorem 1}
For any Coxeter system $(W,S)$ we have
$$
\align
W(t,q) &=\sum_{T \subseteq S} {t^{|T|}
(1-t)^{|S-T|}} \,\, {W(q) \over W_{S-T}(q)} \tag2 \\
{ W(t,q) \over (1-t)^{|S|}} &=
\sum_{ T \in S^\nonnegs} t^{|T|} {W(q) \over W_{S-\hat{T}}(q)} \tag3
\endalign
$$
\endproclaim
\demo{Proof}
We prove equation (2), from which (3) follows easily.
Starting with the right-hand side of (2), one has
$$
\aligned
& \sum_{T \subseteq S} {t^{|T|}
(1-t)^{|S-T|}} \,\, {W(q) \over W_{S-T}(q)} \\
&=\sum_{T \subseteq S} {t^{|T|}
(1-t)^{|S-T|}} \,\, \sum_{w \in W:\Des(w) \subseteq T} q^{l(w)} \\
&=\sum_{w \in W} q^{l(w)} \sum_{\Des(w) \subseteq T \subseteq S}
t^{|T|} (1-t)^{|S-T|} \\
&=\sum_{w \in W} q^{l(w)} t^{\des(w)}
\sum_{\varnothing \subseteq T' \subseteq S-\Des(w)}
t^{|T'|} (1-t)^{|S-\Des(w)-T'|} \\
&=\sum_{w \in W} q^{l(w)} t^{\des(w)}
(t+(1-t))^{|S-\Des(w)|} \\
&=\sum_{w \in W} q^{l(w)} t^{\des(w)}\\
&=W(t,q) \blacksquare
\endaligned
$$
\enddemo
\remark{Remarks}
The specialization of equation (2) to $q=1$ appears as
\cite{Ste, Proposition 2.2(b)}, and the special case of (2)
in which $W$ is of type $A_n$ appears
in slightly different form as \cite{DF, equation (2.5)}.
It is just as easy to refine equations (2), (3) to keep track of
the entire descent set $\Des(w)$ by giving each $s \in S$ its own
indeterminate $t_s$.
One can also refine this computation to incorporate other statistics
than the length function $l(w)$, as long as the statistic
$n(w)$ in question is {\it additive} under every parabolic coset
decomposition in the following
sense: for all $J \subseteq S$,
when $w \in W$ is written uniquely as $w = u \cdot v$ with
$u \in W^J, v \in W_J$, we have $n(w)=n(u)+n(v)$. The following
theorem is then proven in exactly the same fashion as Theorem 1:
\proclaim{Theorem 1$^\prime$}
Let $(W,S)$ be a Coxeter system, and $n_1(w), n_2(w), \ldots$
a series of additive statistics. Then using the notations
$$
\align
{\bold q}^{{\bold n}(w)} &= \prod_i q_i^{n_i(w)} \\
{\bold t}^{T} &= \prod_{s \in T} t_s \\
(1-{\bold t})^{T} &= \prod_{s \in T} (1-t_s)\\
W(\bold q) &= \sum_{w \in W} {\bold q}^{{\bold n}(w)} \\
W(\bold t, \bold q) &= \sum_{w \in W} {\bold t}^{\Des(w)} {\bold q}^{{\bold n}(w)}
\endalign
$$
we have
$$
\align
W(\bold t, \bold q) &=\sum_{subset \,\, T \subseteq S} {{\bold t}^{T}
(1-{\bold t})^{S-T}} \,\, {W({\bold q}) \over W_{S-T}(\bold q)}\tag4\\
{ W(\bold t, \bold q) \over (1-{\bold t})^{S}} &=
\sum_{T \in S^\nonnegs} {\bold t}^T
{W(\bold q) \over W_{S-\hat{T}}(\bold q)} \blacksquare \tag5
\endalign
$$
\endproclaim
In light of this theorem, it is useful to know a classification
of the additive statistics on $W$:
\proclaim{Proposition 2}
Let $(W,S)$ be a Coxeter system, and let $n:W \rightarrow \integers$
be an additive statistic in the above sense. Then
\roster
\item"1." The statistic $n$ is completely determined
by its values on $S$ via the
formula
$$
n(w) = \sum_{j=1}^{l(w)} n(s_{i_j})
$$
for any reduced decomposition $w=s_{i_1} s_{i_2} \cdots s_{i_{l(w)}}$.
\item"2." The statistic $n$ is well-defined if and only if it
is constant on the
$W$-conjugacy classes restricted to $S$, which are well-known (see e.g.
\cite{Hu, Exercise \S 5.3}) to coincide with the connected components of
nodes in the subgraph induced by the odd-labelled edges of the Coxeter diagram.
\endroster
As a consequence, there is a {\it universal} tuple of additive statistics
$n_1, n_2, \ldots$ whose multivariate distribution specializes to that
of any other additive statistics, defined by setting $n_i \vert_S$ to be
the characteristic function of the $i^{th}$ $W$-conjugacy class restricted
to $S$.
\endproclaim
\demo{Proof}
If $n$ is additive, then the decomposition $1=1 \cdot 1$
implies $n(1)=n(1)+n(1)$ so $n(1)=0$.
If the values of $n$ on $S$ are specified, then $n(w)$ is
determined by the formula in the proposition for any $w$,
using induction on $l(w)$: choose any $s \in \Des(w)$, and
then $w = ws \cdot s$ is the unique decomposition in
$W^{\{s\}} \cdot W_{\{s\}}$, so $n(w)=n(ws)+n(s)$.
To prove the second assertion, note that if $s, s'$ are connected
by an odd-labelled edge in the Coxeter diagram, then the longest
element of $W_{\{s,s'\}}$ has two reduced decompositions
$$
s \, s' s \, \cdots = s' \, s \, s' \, \cdots
$$
and the formula for $n$ forces $n(s) = n(s')$.
So $n$ must be constant on the $W$-conjugacy
classes restricted to $S$, and Tits' solution to the word problem for
$(W,S)$ \cite{Hu, \S 8.1} shows that any such function on $S$ will
extend (by the above formula) to a well-defined additive function on $W$.
$\blacksquare$
\enddemo
\endremark
Recall \cite{Hu, \S 1.11, \S 5.12}
the fact that $W(q)$ is a rational function
in $q$, which may be computed using the recursion
$$
W(q) = f(q) \left(
\sum_{J \subsetneq S} {(-1)^{|J|} \over W_J(q)}
\right)^{-1}
\tag6
$$
where
$$
f(q) = \left\{ \matrix (-1)^{|S|+1} &\text{ if }W\text{ is infinite}\\
q^{l(w_0)}+(-1)^{|S|+1}
&\text{ if }W\text{ is finite} \endmatrix
\right.
$$
and $w_0$ is the element of maximal length in $W$ when $W$ is finite.
From equation (2), we conclude that $W(t,q)$ is also a rational
function in $t$ and $q$ (in fact a polynomial in $t$ with coefficients
given by rational functions of $q$, i.e. $W(t,q) \in \integers(q)[t]$).
More generally, the $\bold q$-analogue of recursion (6)
in which $q$ is replaced by $\bold q$ and $l(w)$ by ${\bold a}(w)$
follows from the same proof as (6).
Therefore $W(\bold q) \in \integers(\bold q)$ for any additive
statistics $a_1(w),a_2(w),\ldots$, and from equation (4) we conclude that
$W(\bold t,\bold q) \in \integers(\bold q)[\bold t]$.
Before leaving this folklore section,
we note a happy occurrence when the Coxeter diagram
for $W$ is {\it linear}, i.e. when it
has no nodes of degree greater than or equal to 3.
In this situation and with $q=1$, Stembridge
\cite{Ste, Proposition 2.3, Remark 2.4} observed
that the right-hand side of (2) has a concise determinantal
expression, and the proof given there
generalizes in a straightforward fashion to
prove the following:
\proclaim{Theorem 3}
Let $(W,S)$ be a Coxeter system with linear Coxeter diagram, and
label the nodes $1,2,\ldots,n$ in linear order.
Then
$$
W(\bold t, \bold q) = W(\bold q) \,\, det[a_{ij}]_{0 \leq i,j \leq n}
$$
where
$$
a_{ij}= \left\{
\matrix 0 & i-j>1 \\
t_i-1 & i-j=1 \\
{t_i \over W_{[i+1,j]}(\bold q)} & i \leq j
\endmatrix \right.
$$
and by convention $t_0=1$, and $W_{[i+1,i]}$ is the trivial group
with $1$ element.$\blacksquare$
\endproclaim
For example, if $W$ is the Weyl group of type $B_n (= C_n)$,
then the Coxeter diagram is a path with $n$ nodes having
all edges labelled 3 except for one on the end labelled 4.
An interesting additive statistic $n(w)$ is the
number of times the Coxeter generator on the end with
the edge labelled 4 occurs in a reduced word for $w$
(this is the same as the number of negative signs occurring
in $w$ when considered as
a {\it signed permutation}). It is not hard to check (see e.g.
\cite{Re1, Lemma 3.1}) that if we let
${\bold q}^{{\bold n}(w)} = a^{n(w)} q^{l(w)}$,
then
$$
B_n(\bold q) = (-aq;q)_n [n]!_q
$$
and hence the above determinant is very explicit.
For example when $n=2$,
$$
\align
B_2(\bold t, \bold q)
&= (-aq;q)_2 \, [2]!_q \,\,det \left[ \matrix
1 & {1 \over [2]!_q} & {1 \over (-aq;q)_2 [2]!_q} \\
t_1-1 & t_1 & {t_1 \over (-aq;q)_1 [1]!_q}\\
0 & t_2-1 & t_2
\endmatrix \right] \\
&= 1 + q t_1 + a q^2 t_1 + a q^3 t_1 + a q t_2 + a q^2 t_2 +
a^2 q^3 t_2 + a^2 q^4 t_1 t_2.
\endalign
$$
\subheading{III. $W(t,q)$ for infinite families}
In this section we use equation (2) to compute
the generating function encompassing $W^{(n)}(t,q)$ for all $n$,
where $W^{(n)}$ is an infinite family of Coxeter groups which
grows in a certain prescribed fashion. It turns out that all
of the infinite families of finite and affine Coxeter groups
fit this description, and we deduce generating functions for
their $q$-Eulerian polynomials (and some more general
infinite families) as corollaries.
We begin by describing the infinite family $W^{(n)}$.
Let $(W,S)$ be a Coxeter
system, and choose a particular generator $v \in S$ to distinguish.
Partition the neighbors of $v$ in the Coxeter diagram for
$(W,S)$ into two blocks $B_1, B_2$, and define
$(W^{(n)},S^{(n)})$ for $n \in \nonnegs$ to be the Coxeter system
whose diagram is obtained from that of $(W,S)$ as follows:
replace the node $v$ with a path having $n+1$ vertices $s_0,\ldots,s_n$
and $n$ edges all labelled 3, then connect $s_0$ to the elements of $B_1$
using the same edge labels as $v$ used, and similarly connect $s_n$
to the elements of $B_2$. For example,
$(W^{(0)},S^{(0)}) = (W,S)$, while $(W^{(1)},S^{(1)})$ will have one
more node and one more edge (labelled 3) in its diagram than $(W,S)$ had.
The goal of this section is to compute an expression for the generating
function
$$
\sum_{n \geq 0} {x^n \over W^{(n)}(q)} W^{(n)}(t,q)
$$
For a subset $J \subseteq S-v$, let $(W_J^{(n)},S_J^{(n)})$ be the
Coxeter system corresponding to the parabolic subgroup generated by
$J \cup \{s_0,\ldots,s_n\}$. Also define for $J \subseteq S-v$
and $a,b \in \nonnegs$ the Coxeter system $(W_J^{(a,b)},S_J^{(a,b)})$
to be the one corresponding to the parabolic subgroup of
$(W^{(a+b)},S^{(a+b)})$ generated by $J \cup (\{s_0,\ldots,s_n\}-s_a)$.
Let
$$
\aligned
\exp_{W_J}(x;q) &= \sum_{n \geq 0} {x^n \over W_J^{(n)}(q)} \\
\dex_{W_J}(x;q) &= \sum_{a,b \geq 0} {x^{a+b} \over W_J^{(a,b)}(q)}
\endaligned
$$
The terminologies ``$\exp$" and ``$\dex$" are intended to be suggestive
of the fact that in the special cases of interest, $\exp_{W_J}(x;q)$
will be related to a $q$-analogue of the exponential function $\exp(x)$,
and $\dex_{W_J}(x;q)$ will either be a product of two
such $q$-analogues of exponentials (so a {\it double} exponential)
or the {\it derivative} of such a $q$-analogue.
\proclaim{Theorem 4}
$$
\align
&\sum_{n \geq 0} {x^n \over W^{(n)}(q)} W^{(n)}(t,q) =\\
&\left[
\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|}
\left( \exp_{W_{S-v-J}}(x;q) +
{t \, \dex_{W_{S-v-J}}(x;q) \over 1- t\, \exp(x;q)} \right)
\right]_{x \mapsto x(1-t)}
\endalign
$$
\endproclaim
\demo{Proof}
From equation (2) we have
$$
W^{(n)}(t,q) =\sum_{T \subseteq S^{(n)}}
t^{|T|} (1-t)^{|S^{(n)}-T|}
\,\, { W^{(n)}(q) \over W^{(n)}_{S^{(n)}-T}(q) }
$$
so that
$$
\aligned
&{W^{(n)}(t,q) \over W^{(n)}(q) \,\, (1-t)^n} \\
&=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|}
\sum_{K \subseteq \{s_0,\ldots,s_n\} } {t^{|K|}\over (1-t)^{|K|}}
\,\, {1 \over W^{(n)}_{S^{(n)}-J-K}(q)} \\
&=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|}
\sum_{K \in \nonnegs^{\{s_0,\ldots,s_n\}} } t^{|K|}
\,\, {1 \over W^{(n)}_{S^{(n)}-J-\hat{K}}(q)} \\
&=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|}
\left( {1 \over W^{(n)}_{S-v-J}(q)} +
\sum_{k \geq 1} t^k
\sum_{K \in \nonnegs^{\{s_0,\ldots,s_n\}} \atop |K|=k }
\,\, {1 \over W^{(n)}_{S^{(n)}-J-\hat{K}}(q)} \right)
\endaligned
$$
At this stage, we use an encoding for the functions
$K:\{s_0,\ldots,s_n\} \rightarrow \nonnegs$ having $|K|=k$.
Let $\omega_i \in \reals^n$
be the vector $e_1+e_2+\ldots+e_i$, where $e_i$ is the $i^{th}$ standard
basis vector, so that $\omega_0=(0,0,\ldots,0)$ and $\omega_n=(1,1,\dots,1)$.
Given $K:\{s_0,\ldots,s_n\}~\rightarrow~\nonnegs$, encode it as the
vector $c(K) = \sum_{i=0}^n K(s_i) \, \omega_i \in \reals^n$. Note
that once we have fixed the cardinality $|K|=k \geq 1$, then $K$ is completely
determined by $c(K)$, which is a decreasing sequence with entries
in the range $[0,k]$. Hence $K$ is also completely determined
by the sequence $a(K)=(a_0,\ldots,a_k)$ where $a_i$ is the number
of occurrences of $i$ in $c(K)$. Furthermore, it is easy to check that
the parabolic subgroup $W_{S^{(n)}-J-\hat{K}}$ is then isomorphic to
$$
W_{S-v-J}^{(a,b)} \times S_{a_1} \times \cdots \times S_{a_{k-1}}.
$$
Therefore we may continue the calculation
$$
\aligned
&{W^{(n)}(t,q) \over W^{(n)}(q) \,\, (1-t)^n}
=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|} \times \\
& \left( {1 \over W^{(n)}_{S-v-J}(q)} +
\sum_{k \geq 1} t^k
\sum_{(a_0,\ldots,a_k) \in \nonnegs^{k+1} \atop \sum a_i=n }
\,\, {1 \over W_{S-v-J}^{(a,b)}(q) \, [a_1]!_q \cdots [a_{k-1}]!_q } \right) \\
&\sum_{n \geq 0}{W^{(n)}(t,q) \over W^{(n)}(q)} \,\,{x^n \over (1-t)^n}
=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|} \times \\
& \left( \sum_{n \geq 0} {x^n \over W^{(n)}_{S-v-J}(q)} +
\sum_{k \geq 1} t^k \sum_{n \geq 0}
\sum_{(a_0,\ldots,a_k) \in \nonnegs^{k+1} \atop \sum a_i=n }
\,\, {x^{a_0+a_k} \over W_{S-v-J}^{(a_0,a_k)}(q)}
{x^{a_1} \over [a_1]!_q} \cdots {x^{a_{k-1}} \over [a_{k-1}]!_q}
\right) \\
&=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|} \times \\
& \left( \exp_{W_{S-v-J}}(x;q) +
\sum_{a_0,a_k \geq 0} {x^{a_0+a_k} \over W_{S-v-J}^{(a,b)}(q)}
\sum_{k \geq 1} t^k (\exp(x;q))^k \right)\\
&=\sum_{J \subseteq S-v} t^{|J|} (1-t)^{|S-J|}
\left( \exp_{W_{S-v-J}}(x;q) +
\dex_{W_{S-v-J}}(x;q) {t \over 1-t\,\exp(x;q)} \right)
\endaligned
$$
The theorem now follows upon replacing $x$ by $x(1-t)$.$\blacksquare$
\enddemo
\remark{Remarks}
\roster
\item "1." The crucial encoding of functions
$K:\{s_0,\ldots,s_n\} \rightarrow \nonnegs$
used in the middle of the preceding proof is a translation
and generalization of the ``direct encoding"
used in \cite{GG, \S1} for type $A_n$.
\item "2." There is an obvious ${\bold q}$-analogue of
Theorem 3 involving additive statistics on $(W,S)$, with
the same proof.
\endroster
\endremark
\subheading{IV. Explicit generating functions for classical
Weyl groups and affine Weyl groups}
This section (and the remainder of the paper)
is devoted to specializing Theorem 4 to
compute generating functions for descents and length in
all of the classical finite and affine Weyl groups, and
certain families which generalize them.
In all cases where $W$ is a finite or affine Weyl group,
the denominators $W(q)$ occurring in the left-hand side
of Theorem 4 can be made explicit for the following reason:
if $W$ is a finite {\it Weyl}
group of rank $n$, then there is an associated multiset of
numbers $e_1,e_2,\ldots,e_n$ called the {\it exponents} of $W$,
satisfying
$$
\align
W(q) &= \prod_{i=1}^n [e_i+1]_q \tag7 \\
\tilde{W}(q) &= \prod_{i=1}^n {[e_i+1]_q \over 1-q^{e_i}} \tag8
\endalign
$$
where $\tilde{W}$ is the affine Weyl group associated to $W$.
The first formula is a theorem of Chevalley \cite{Hu, \S 3.15},
the second a theorem of Bott \cite{Hu, \S 8.9}. We should mention
that Bott's proof, although extremely elegant and
unified, is not completely elementary, and more elementary proofs of some
cases of his theorem have recently appeared in \cite{BB, BE , EE, ER}.
We first consider an infinite family of Coxeter systems with
{\it linear} diagrams. Let $W^{r,s}_n$ be the family of Coxeter
groups whose Coxeter diagram is a path with $n$ nodes,
in which the labels on almost all of the edges are 3
{\it except} for the leftmost edge labelled $r$ and the rightmost
edge labelled $s$. Let $W^{r}_n$ be the family defined by
$W^{r}_n=W^{r,3}_n$ The next result uses Theorem 4 to compute a generating
function for $W^{r,s}_n(t,q)$. Note
that $W^{r,s}_n$ contains as special cases the finite Coxeter groups of type
$A_n, B_n (=C_n), H_3, H_4$, and the affine Weyl groups $\affC_n$,
as well as some hyperbolic Coxeter groups (see \cite{Hu, \S 2.4, 2.5, 6.9}).
Before stating the theorem, we establish some more notation.
Let
$$
\aligned
\exp_{W^r}(x;q) & = \sum_{n \geq 0} {x^n \over W^r_n(q)} \\
\exp_{W^{r,s}}(x;q) & = \sum_{n \geq 0} {x^n \over W^{r,s}_n(q)}
\endaligned
$$
where by convention we define
$W^{r,s}_0=W^r_0$ to be the trivial group
with 1 element, $W^{r,s}_1=W^r_1$ is the unique Coxeter system
of rank $1$, and $W^{r,s}_2=W^r_2= I_2(r)$ is the rank 2 (dihedral)
Coxeter system of order $2r$.
\proclaim{Theorem 5}
$$
\align
\sum_{n \geq 0}{x^n \over W^{r,s}_n(q)} W^{r,s}_n(t,q)
& = \exp_{W^{r,s}}(x(1-t);q) \tag9\\
& + {t \, x \, (1-t) \, \exp_{W^r}(x(1-t);q) \, \exp_{W^s}(x(1-t);q) \over
1 - t \, \exp(x(1-t);q)} \\
\sum_{n \geq 0}{x^n \over W^{r}_n(q)} W^{r}_n(t,q)
& = {(1-t) \,\exp_{W^{r}}(x(1-t);q) \over 1 - t \, \exp(x(1-t);q)}
\tag10
\endalign
$$
\endproclaim
\demo{Proof}
Equation (10) follows from equation (9) by setting
$s=3$ and noting that
$$
\align
\exp_{W^{r,3}}(x;q) &= \exp_{W^r}(x;q) \\
\exp_{W^3}(x;q) &= {\exp(x;q) - 1 \over x}.
\endalign
$$
We wish to derive equation (9) from Theorem 4.
In the notation preceding Theorem 4,
choose $(W,S)$ to have Coxeter diagram with 3 nodes
$s_1,s_2,s_3$ forming a path with two edges $\{s_1, s_2\}, \{s_2, s_3\}$
labelled $r$ and $s$ respectively, and let $v=s_2,
B_1=\{s_1\}, B_2=\{s_2\}$. One can then check
that
$$
\aligned
W^{(n)} &= W^{r,s}_{n+3}\\
\exp_{W_{s_1,s_3}}(x;q) &=
x^{-3}\left(\exp_{W^{r,s}}(x;q)
-1-{x \over [2]_q}-{x^2 \over [2]_q [r]_q}\right) \\
\exp_{W_{s_1}}(x;q) &=x^{-2}\left(\exp_{W^r}(x;q)-1-{x \over [2]_q}\right) \\
\exp_{W_{s_3}}(x;q) &=x^{-2}\left(\exp_{W^s}(x;q)-1-{x \over [2]_q}\right) \\
\exp_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1-x) \\
\dex_{W_{s_1,s_3}}(x;q) &=
x^{-2}(\exp_{W^r}(x;q)-1)(\exp_{W^s}(x;q)-1) \\
\dex_{W_{s_1}}(x;q) &=x^{-2}(\exp_{W^r}(x;q)-1)(\exp(x;q)-1)) \\
\dex_{W_{s_3}}(x;q) &=x^{-2}(\exp_{W^s}(x;q)-1)(\exp(x;q)-1)) \\
\dex_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1)^2
\endaligned
$$
and using these facts, equation (9) follows from Theorem 4 with a little
algebra.
$\blacksquare$
\enddemo
\noindent
We now specialize Theorem 5 to obtain generating functions for
the types $A_{n-1} (=S_n), B_n (=C_n)$, and $\affC_n$.
If $r=3$ then $W^r_{n}$ coincides with the finite Weyl group
$A_n(=S_{n+1})$ which has exponents $1,\ldots,n$, and one can check
that equation (10) is equivalent to Stanley's formula (1).
It is interesting to note that $\exp_{W^r}(x;q)$ has an alternate
expression in this case in terms of an infinite product, since
$\exp_{W^r}(x;q) = x^{-1}(\exp(x;q)-1)$ as noted earlier, and
$$
\exp(x;q) = \sum_{n \geq 0} {(x(1-q))^n \over (q;q)_n} = (x(1-q);q)_\infty^{-1}
$$
where the last equality is by the $q$-binomial theorem \cite{GR, Appendix II.3}:
$$
\sum_{n \geq 0} {(z;q)_n \over (q;q)_n} x^n =
{(zx;q)_\infty \over (x;q)_\infty}.
$$
If $r=4$ then $W^r_{n}$ coincides with the finite Weyl group
$B_n$ or $C_n$ which has exponents $1,3,\ldots,2n-1$.
In this case equation (10) is equivalent to \cite{Re1, \S3} specialized
to $a=q=1$. Again we note that $\exp_{W^r}(x;q)$ has an alternate
expression in this case as an infinite product, since
$$
\exp_{W^4}(x;q) = \sum_{n \geq 0} {(x(1-q))^n \over (q^2;q^2)_n} = (x(1-q);q^2)_\infty^{-1}
$$
again by the $q$-binomial theorem. Furthermore,
since the Coxeter diagram in
the case has an edge labelled $4$, there exists another additive
statistic $n(w)$, equal to the number of negative signs in
$w$ considered as a signed permutation (see example after Theorem 3).
Using the known distribution
$$
B_n(\bold q) = \sum_{w \in B_n} a^{n(w)} q^{l(w)} = (-aq;q)_n [n]!_q
$$
the proof of Theorem 3 for $r=4$ can be refined to a result
equivalent to \cite{Re1, \S3, specialized to $q=1$}.
On the other hand, it does not seem to be true
that the generalization of $\exp_{W^4}(x;q)$
defined by
$$
\exp_{W^4}(x;a,q) = \sum_{n \geq 0} {x^n \over (-aq;q)_n [n]!_q}
$$
has a nice infinite product expression.
If $r=s=4$ then $W^{r,s}_{n+1}$ coincides with the affine Weyl group
$\affC_{n}$ for $n \geq 2$, so equation (9) says
$$
\align
& 1 + {x (1+tq) \over [2]_q}+{x^2 C_2(t,q) \over [2]_q [4]_q}
+ x \, \sum_{n \geq 2}{x^n \over \affC_n(q)} \affC_n(t,q) \\
& =
\exp_{W^{4,4}}(x(1-t);q) +
{t \, x \, (1-t) \, \left[ \exp_{W^4}(x(1-t);q) \right] ^2 \over
1 - t \, \exp(x(1-t);q)}.
\endalign
$$
Again we can replace $\exp(x;q), \exp_{W^4}(x;q)$ by their
infinite product formulas as before, and $\exp_{W^{4,4}}(x;q)$
also has an expression involving an infinite product:
since the associated Weyl group $C_n$ has exponents $1,3, \ldots, 2n-1$,
by (8) we have
$$
W^{4,4}_{n+1} =\affC_{n}(q) = {(q^2;q^2)_n \over (1-q)^n (q;q^2)_n}
\tag11
$$
for $n \geq 2$ and hence
$$
\align
\exp_{W^{4,4}}(x;q) & = 1 + {x \over [2]_q}+{x^2 \over [2]_q [4]_q} +
x \, \sum_{n \geq 2} {(q;q^2)_n \over (q^2;q^2)_n} (x(1-q))^n \\
& = 1 + {x \over [2]_q}+{x^2 \over [2]_q [4]_q} +
x \, \left( {(xq(1-q);q^2)_\infty \over (x(1-q);q^2)_\infty}
- {x(1-q)\over 1+q} - 1 \right)
\endalign
$$
where the last equality comes from the $q$-binomial theorem.
Furthermore, since the Coxeter diagram in this case has its two
extreme edges labelled $4$, there exist
two other additive statistic $n(w), m(w)$ equal to the number
of occurrences of the two endpoint Coxeter generators occurring
in a reduced word for $w$.
One can prove the following refinement of equation (11)
(a special case of Bott's Theorem) for $W_{n+1}^{4,4}=\affC_n$:
$$
\align
\affC_n(\bold q) & = \sum_{w \in \affC_n} a^{n(w)} b^{m(w)} q^{l(w)} \\
& = {(-aq;q)_n (-bq;q)_n [n]!_q \over (abq^{n+1};q)_n} \tag12
\endalign
$$
by using the ${\bold q}$-generalization of recursion (6) to show
$$
\align
{1 \over \affC_n(\bold q)} & = \sum_{i=0}^n
{q^{i^2} a^i \over B_i(a,q) B_{n-i}(b,q)} \\
& = \sum_{i=0}^n
{q^{i^2} a^i \over (-aq;q)_i [i]!_q (-bq;q)_{n-i} [n-i]!_q}
\endalign
$$
and then applying the $q$-Vandermonde summation formula
\cite{GR, Appendix II.6}.
The ${\bold q}$-refinement of Theorem 5 with $r=s=4$
then gives a very explicit generating function generalization enumerating
$\affC_n$ by the quadruple of statistics
$$
(n(w), m(w), l(w), \des(w)).
$$
On the other hand, it no longer seems to be true that there is a nice
infinite product expression for the
relevant generalization of $\exp_{W^{4,4}}(x;q)$ defined by
$$
\align
\exp_{W^{4,4}}(x;a,b,q) & = \sum_{n \geq 0}
{(abq^{n+1};q)_n \over (-bq;q)_n (-aq;q)_n [n]!_q} x^n \\
&= {}_4\phi_3 \left(
\left. \matrix \sqrt{abq} & -\sqrt{abq} &\sqrt{ab}q &-\sqrt{ab}q \\
-aq & -bq & abq &
\endmatrix \right\vert x(1-q);q \right)
\endalign
$$
where the last equation is basic hypergeometric
series notation (see e.g. \cite{GR}).
We next deal with the affine symmetric groups.
Let $\tilde{A}_{n-1} = \tilde{S}_{n}$ be the affine Weyl group
corresponding to the Weyl group $A_{n-1} =S_{n}$,
so $\affS_n$ has as its Coxeter diagram a cycle with $n$ vertices
and label 3 on every edge. Let
$$
\exp_{\affS}(x:q)=\sum_{n \geq 1} {x^n \over \affS_n(q)}.
$$
We now prove a formula claimed in the Introduction:
\proclaim{Theorem 6}
$$
\sum_{n \geq 1} { x^n \over 1-q^n } \affS_n(t,q)
= \left[{ x {\partial \over \partial x} \log(\exp(x;q))
\over 1-t\, \exp(x;q) }
\right]_{x \mapsto x\,{1-t \over 1-q}}.
$$
\endproclaim
\demo{Proof}
In the notation preceding Theorem 4,
choose $(W,S)$ to have Coxeter diagram with 3 nodes
$s_1,s_2,s_3$ arranged in a triangle with the three edges labelled 3.
Let $v=s_3$ and $B_1=\{s_1\}, B_2=\{s_2\}$.
One can then check that
$$
\aligned
W^{(n)} &= \affS_{n+3}\\
\exp_{W_{s_1,s_2}}(x;q)
&= x^{-3}\left(\exp_{\affS}(x;q)-x-{x^2 (1-q) \over [2]_q}\right) \\
\exp_{W_{s_1}}(x;q) &= \exp_{W_{s_2}}(x;q)
= x^{-3}\left(\exp(x;q) 1-x-{x^2 \over [2]_q}\right) \\
\exp_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1-x) \\
\dex_{W_{s_1,s_2}}(x;q)
&= -2x^{-3}\left(\exp(x;q)-1-x-{x^2 \over [2]_q}\right) \\
& +x^{-2}\left({\partial \over \partial x} \exp(x;q)
-1-{2x \over [2]_q}\right) \\
\dex_{W_{s_1}}(x;q) &=\dex_{W_{s_2}}(x;q)
= x^{-3} (\exp(x;q)-1) (\exp(x;q)-1-x) \\
\dex_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1)^2
\endaligned
$$
and using these facts one can simplify Theorem 4 in this case to
$$
\sum_{n \geq 1} { x^n \over \affS_n(q) } \affS_n(t,q)
= \left[\exp_{\affS}(x;q) +
{t \,x \, {\partial \over \partial x}
\exp(x;q) \over 1 - t \, \exp(x;q)} \right]_{x \mapsto x\,(1-t)}.
$$
To rewrite this more explicitly, we note that
the exponents of $S_n = A_{n-1}$ are $1,2,\ldots,n-1$,
so that equation (8) gives
$$
\affS_n(q) = {[n]!_q \over (q;q)_{n-1}} = {1-q^n \over (1-q)^n}.
$$
Therefore
$$
\aligned
\exp_{\affS}(x;q)
&= \sum_{n \geq 1} {x^n (1-q)^n \over 1-q^n} \\
&= \sum_{n \geq 1} \sum_{m \geq 0} (x(1-q))^n q^{nm} \\
&= \sum_{m \geq 0} \sum_{n \geq 1}(x(1-q)q^m)^n \\
&= \sum_{m \geq 0} {x(1-q)\,q^m \over 1-x(1-q)\,q^m} \\
&= \sum_{m \geq 0} x {\partial \over \partial x}
\log[(1-x(1-q)\,q^m)^{-1}]\\
&= x {\partial \over \partial x} \log[(x(1-q);q)^{-1}_\infty] \\
&= x {\partial \over \partial x} \log(\exp(x;q))
\endaligned
$$
Substituting this into the last equation and replacing $x$ by ${x \over 1-q}$
gives
$$
\sum_{n \geq 1} { x^n \over 1-q^n } \affS_n(t,q)
= \left[ x {\partial \over \partial x} \log(\exp(x;q)) +
{t \,x \, {\partial \over \partial x}
\exp(x;q) \over 1 - t \, \exp(x;q)}
\right]_{x \mapsto x{1-t \over 1-q}}
$$
which is equivalent to the theorem by a little algebra.$\blacksquare$
\enddemo
Next we move on to a common
generalization of the Weyl groups $D_n$ and the
affine Weyl groups $\affB_n$.
Let $D^r_n$ be the Coxeter system whose graph is obtained
from the graph for $D_n$ by replacing the label of $3$ on the edge
farthest from the ``fork" with a label of $r$. Note that $D^3_n = D_n$
and $D^4_n = \affB_n$ (see \cite{Hu, \S 2.4, 2.5}). We adopt the
notation
$$
\align
\exp_{D^r}(x;q) & = \sum_{n \geq 2} {x^n \over D^r_n(q)} \\
\exp_{D}(x;q) &= \sum_{n \geq 2} {x^n \over D_n(q)}
\endalign
$$
where by convention we define $D^r_2 = A_1 \oplus A_1$ and
$D^r_3 = A_3$.
\noindent
{\bf Remark:}
The notation $\exp_{D}(x;q)$ is slightly different from the
notation $\exp_D(u)$ used in \cite{Re2, Corollary 4.5}, and in
fact, there is an error in this previous reference,
which we correct here:
the definition of $\exp_D(u)$ given there as
$$
\exp_D(u) = \sum_{n \geq 0} {u^n \over (-q;q)_{n-1} [n]!_q}
$$
should actually read
$$
\exp_D(u) = 2 + \sum_{n \geq 1} {u^n \over (-q;q)_{n-1} [n]!_q}
$$
Therefore this previous definition of $\exp_D(u)$ differs from
our present notation $\exp_D(u;q)$ in the coefficients of $u^0,u^1$.
\proclaim{Theorem 7}
$$
\split
&\sum_{n \geq 4}{ x^n \over D^r_n(q) } D^r_n(t,q) = \\
&\sum_{n \geq 4} \langle x^n \rangle \left[ \exp_{D^r}(x;q) +
{t \, x \, \exp_{W^r}(x;q)
\over 1 - t \, \exp(x;q)} \left(2-{tx\over 1-t}+\exp_D(x;q)\right)
\right]_{x \mapsto x(1-t)} \cdot x^n
\endsplit \tag13
$$
\endproclaim
\demo{Proof}
In the notation preceding Theorem 4,
choose $(W,S)$ to have Coxeter diagram with 4 nodes
$s_1,s_2,s_3,s_4$ in which $s_4$ is connected by an edge
labelled $3$ to $s_1, s_2$, and connected to $s_3$ by an edge labelled
$r$, with no other edges in the diagram. Let $v=s_4$ and
$B_1=\{s_1,s_2\}, B_2=\{s_3\}$.
One can then check that
$$
\aligned
W^{(n)} &= D^{r}_{n+4}\\
\exp_{W_{s_1,s_2,s_3}}(x;q) &=
x^{-4}\left(\exp_{D^r}(x;q)
-{x^2 \over ([2]_q)^2}-{x^3 \over [3]!_q}\right) \\
\exp_{W_{s_1,s_2}}(x;q) &=
x^{-3}\left(\exp_{D}(x;q)
-{x^2 \over ([2]_q)^2}\right) \\
\exp_{W_{s_1,s_3}}(x;q) &= \exp_{W_{s_2,s_3}}(x;q)
=x^{-3}\left(\exp_{W^r}(x;q)
-1-{x \over [2]!_q}-{x^2 \over [2]_q [r]_q}\right) \\
\exp_{W_{s_1}}(x;q) &= \exp_{W_{s_2}}(x;q)
= x^{-3}\left(\exp(x;q) -1-x - {x^2 \over [2]!_q}\right) \\
\exp_{W_{s_3}}(x;q) &=
x^{-2}\left(\exp_{W^r}(x;q) -1- {x \over [2]!_q}\right) \\
\exp_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1-x) \\
\dex_{W_{s_1,s_2,s_3}}(x;q) &=
x^{-3} \exp_D(x;q) (\exp_{W^r}(x;q)-1) \\
\dex_{W_{s_1,s_2}}(x;q) &=
x^{-3} \exp_D(x;q) (\exp(x;q)-1) \\
\dex_{W_{s_1,s_3}}(x;q) &= \dex_{W_{s_2,s_3}}(x;q)
=x^{-3} (\exp_{W^r}(x;q)-1) (\exp(x;q)-1-x) \\
\dex_{W_{s_1}}(x;q) &= \dex_{W_{s_2}}(x;q)
= x^{-3} (\exp(x;q)-1) (\exp(x;q)-1-x) \\
\dex_{W_{s_3}}(x;q) &=
x^{-2} (\exp_{W^r}(x;q)-1) (\exp(x;q)-1) \\
\dex_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1)^2
\endaligned
$$
and using these the result follows from Theorem 4.
$\blacksquare$
\enddemo
We now specialize Theorem 7 to $r=3, 4$. If $r=3$, then
$D^r_n = D_n$ and then one can check that our conventions for
$D_3$ and $D_2$ have been chosen correctly so that
the generating function on the right-hand side of equation (13)
agrees with the left-hand side in its coefficient of $x^2, x^3$
(as well as $x^n$ for $n \geq 4$).
Therefore we obtain
$$
\split
2tx+\sum_{n \geq 2} & { x^n \over D_n(q) } D_n(t,q) = \\
& {(1-t)\,\exp_{D}(x(1-t);q) + t\,(2-tx)\,(\exp(x(1-t);q)-1)
\over 1-t \, \exp(x(1-t);q)}.
\endsplit
$$
which one can easily check agrees with \cite{Re2, Corollary 4.5}.
Note that since $D_n$ has exponents $1,3,\ldots,2n-3,n-1$
by equation (7) we have
$$
\align
\exp_D(x;q) &= \sum_{n \geq 2} {x^n \over (-q;q)_{n-1} [n]!_q} \\
&= \sum_{n \geq 2} {x^n (1-q)^n (1+q^n) \over (q^2;q^2)_n} \\
&= \sum_{n \geq 2} \left( {(x(1-q))^n \over (q^2;q^2)_n}
+ {(xq(1-q))^n \over (q^2;q^2)_n} \right)\\
&= (x(1-q);q^2)_\infty^{-1} + (xq(1-q);q^2)_\infty^{-1} - 2 - x
\endalign
$$
so one can again replace the exponential functions $\exp(x;q),
\exp_D(x;q)$ appearing above by expressions involving infinite
products if desired.
If $r=4$, then $D^r_n = \affB_{n-1}$, so equation (13) gives
a closed form for the generating function
$$
\sum_{n \geq 3} {x^n \over \affB_n(q)} \affB_n(t,q).
$$
Since $\affB_n$ is the affine Weyl group associated
to $B_n$, which has the same exponents as $C_n$, we must
have $\affB_n(q) = \affC_n(q)$ by equation (7). Therefore
we have already seen how all of the functions
$\exp_{D^4}(x;q), \, \exp_{W^4}(x;q), \, \exp(x;q)$
appearing in the generating function can be made more explicit,
and replaced by expressions involving infinite products if desired.
Furthermore, since the Coxeter diagram for $\affB_n$ has an edge
labelled 4, there is another additive statistic $n(w)$ which counts
how many times the Coxeter generator at that end of the diagram is
used in a reduced word for $w$, and one can derive (similarly to (12))
the following refinement of equation (10) for $W=\affB_n$:
$$
\align
\affB_n(\bold q) & = \sum_{w \in \affB_n} a^{n(w)} q^{l(w)} \\
& = {(-aq;q)_n (-q;q)_{n-1} [n]!_q \over (aq^n;q)_n} \tag14
\endalign
$$
This allows one to refine equation (13) when $r=4$
so as to incorporate the statistic $n(w)$.
However, as before the generalization $\exp_{D^4}(x;a,q)$ of
$\exp_{D^4}(x;q)$ does not seem to have a nice infinite product expression.
The only affine Weyl group remaining to be discussed
is $\affD_n$, whose Coxeter diagram looks like a path with forks
at both ends having $n$ nodes total, and all edges labelled $3$
(see \cite{Hu, \S 2.5}). We use the notation
$$
\exp_{\affD}(x;q)=\sum_{n \geq 4} {x^n \over \affD_n(q)}.
$$
\proclaim{Theorem 8}
$$\align
&\sum_{n \geq 4} {x^n \over \affD_n(q)} \affD_n(t,q) =
\sum_{n \geq 4} \langle x^n \rangle \\
& \left\{(1-t)\,\exp_{\affD}(x(1-t);q)
+ {t \over 1-t\,\exp(x(1-t);q)} \times \right. \\
& \left[
{t^2(2+2x-tx)^2+
t(2+tx)(2-4t-3tx+2t^2x)\,\exp(x(1-t);q)
\over 1-t} \right. \\
& \left.\left.
+2(1-t)(2-tx)\,\exp_D(x(1-t);q)
+(1-t)\,\exp_D(x(1-t);q)^2
\right]\right\}\,x^n
\endalign
$$
\endproclaim
\demo{Proof}
In the notation preceding Theorem 4,
choose $(W,S)$ to have Coxeter diagram with 5 nodes
$s_1,s_2,s_3,s_4,s_5$ in which $s_5$ is connected by an edge
labelled $3$ to $s_1, s_2,s_3,s_4$, and there are no other edges
in the diagram. Let $v=s_5$ and
$B_1=\{s_1,s_2\}, B_2=\{s_3,s_4\}$. One can then check that
$$
\aligned
W^{(n)} &= \affD^{r}_{n+4}\\
\exp_{W_{s_1,s_2,s_3,s_4}}(x;q) &=
x^{-4} exp_{\affD}(x;q) \\
\exp_{W_{s_1,s_2,s_3}}(x;q) &=\exp_{W_{s_1,s_2,s_4}}(x;q)
=\exp_{W_{s_1,s_3,s_4}}(x;q) = \exp_{W_{s_2,s_3,s_4}}(x;q) \\
&=x^{-4}\left(\exp_{D}(x;q)
-{x^2 \over ([2]_q)^2}-{x^3 \over [3]!_q}\right) \\
\exp_{W_{s_1,s_2}}(x;q) &= \exp_{W_{s_3,s_4}}(x;q)
=x^{-3}\left(\exp_D(x;q) -{x^2 \over ([2]_q)^2}\right) \\
\exp_{W_{s_1,s_3}}(x;q) &= \exp_{W_{s_1,s_4}}(x;q)
= \exp_{W_{s_2,s_3}}(x;q)= \exp_{W_{s_2,s_4}}(x;q)\\
&=x^{-4}\left(\exp(x;q)
-1-x-{x^2 \over [2]!_q}-{x^3 \over [3]!_q}\right) \\
\exp_{W_{s_1}}(x;q) &=\exp_{W_{s_2}}(x;q)
=\exp_{W_{s_3}}(x;q) =\exp_{W_{s_4}}(x;q) \\
&= x^{-3}\left(\exp(x;q) -1-x-{x^2 \over [2]!_q}\right) \\
\exp_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1-x) \\
\dex_{W_{s_1,s_2,s_3,s_4}}(x;q) &= x^{-4} \exp_{\affD(x;q)}\\
\dex_{W_{s_1,s_2,s_3}}(x;q) &= \dex_{W_{s_1,s_2,s_4}}(x;q)
=\dex_{W_{s_1,s_3,s_4}}(x;q) = \dex_{W_{s_2,s_3,s_4}}(x;q) \\
&= x^{-4} \exp_D(x;q) (\exp(x;q)-1-x) \\
\dex_{W_{s_1,s_2}}(x;q) &= \dex_{W_{s_3,s_4}}(x;q)
=x^{-3} \exp_D(x;q) (\exp(x;q)-1) \\
\dex_{W_{s_1,s_3}}(x;q) &= \dex_{W_{s_1,s_4}}(x;q)
= \dex_{W_{s_2,s_3}}(x;q)= \dex_{W_{s_2,s_4}}(x;q)\\
&=x^{-4} (\exp(x;q)-1-x)^2 \\
\dex_{W_{s_1}}(x;q) &= \dex_{W_{s_2}}(x;q)
= \dex_{W_{s_3}}(x;q) = \dex_{W_{s_4}}(x;q) \\
&= x^{-3} (\exp(x;q)-1) (\exp(x;q)-1-x) \\
\dex_{W_\varnothing}(x;q) &=x^{-2}(\exp(x;q)-1)^2
\endaligned
$$
and using these facts, the result follows from Theorem 4 with a little algebra.
$\blacksquare$
\enddemo
As in the previous cases of affine Weyl groups, it is possible to
replace $\exp_{\affD}(x)$ by an expression involving infinite products,
if desired. Since $D_n$ has exponents $1,3,5,\ldots,2n-5,2n-3,n-1$,
by equation (8) we have
$$
\align
\affD_n(q) &={(-q;q)_{n-1} [n]!_q \over (1-q)^n (q;q^2)_{n-1} (1-q^{n-1})}\\
&= {(q^2;q^2)_n \over(1-q)^n (q^{-1};q^2)_n} {(1-q^{-1})
\over (1+q^n) (1-q^{n-1}) }\\
\endalign
$$
Therefore
$$
\aligned
\exp_{\affD}(x;q)
&= \sum_{n \geq 4} (x(1-q))^n {(q^{-1};q^2)_n \over (q^2;q^2)_n}
{(1+q^n)(1-q^{n-1}) \over (1-q^{-1})}\\
&= \sum_{n \geq 4} (x(1-q))^n {(q^{-1};q^2)_n \over (q^2;q^2)_n}
\left[(1+q^n) + {1 \over q-1} (1-q^{2n})\right] \\
&= {(xq^{-1}(1-q);q^2)_\infty \over (x(1-q);q^2)_\infty}
+ {(x(1-q);q^2)_\infty \over (xq(1-q);q^2)_\infty} \\
& + xq^{-1}(1-q){(xq(1-q);q^2)_\infty \over (x(1-q);q^2)_\infty}
-\sum_{i=0}^{3} c_i(q) \, x^i \\
&= {(xq(1-q);q^2)_\infty \over (x(1-q);q^2)_\infty}
+ {(x(1-q);q^2)_\infty \over (xq(1-q);q^2)_\infty}
-\sum_{i=0}^{3} c_i(q) \, x^i
\endaligned
$$
for some $c_i(q)$ which are rational functions of $q$.
\subheading{Acknowledgements}
The author would like to thank Anders Bj\"orner, Dennis Stanton,
and John Stembridge for helpful comments, references, and suggestions.
He would also like to thank the referee for helpful suggestions
about clarifying the presentation.
\Refs
\widestnumber\key{Re1}
\ref
\key Br
\by F. Brenti
\paper $q$-Eulerian polynomials arising from Coxeter groups
\jour Europ. J. Combin.
\yr 1994
\vol 15
\pages 417-441
\endref
\ref
\key BB
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\enddocument