THE DISTRIBUTION OF DESCENTS

AND LENGTH IN A COXETER GROUP

Victor Reiner

University of Minnesota

                      e-mail: reiner@math.umn.edu

Submitted August 19, 1995; accepted November 25, 1995

Abstract. We give a method for computing the q-Eulerian distribution

W (t, q) = ∑_{w∈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 length function in W, and des(w) is the number of simple reflections s ∈ S for which l(ws) < l(w). Using this we compute generating functions encompassing the q-Eulerian distributions of the classical infinite families of finite and affine Weyl groups.

I. Introduction.

Let (W, S) be a Coxeter system (see [Hu] for definitions and terminology). There are two statistics on elements of the Coxeter group W

l(w) = min{l : w = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{l}} for some s_{i_{k}} ∈ S}
des(w) = |{s ∈ S : l(ws) < l(w)}|

which generalize the well-known permutation statistics inversion number and descent number in the case W is the symmetric group S_{n}. The polynomial

∑_{w∈S_{n}} t^{des(w)}

is known in the combinatorial literature as the Eulerian polynomial, which has generating function

∑_{n≥0} \frac{x^{n}}{n!} ∑_{w∈S_{n}} t^{des(w)} = \frac{(1 − t) e^{x(1−t)}}{1 − t e^{x(1−t)}}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 05A15, 33C80.

Work supported by Mathematical Sciences Postdoctoral Research Fellowship DMS-9206371

and a q-analogue first computed by Stanley [St, §3]:

(1)  ∑_{n≥0} \frac{x^{n}}{[n]!_{q}} ∑_{w∈S_{n}} t^{des(w)}q^{l(w)} = \frac{(1 − t) exp(x(1 − t); q)}{1 − t exp(x(1 − t); q)}

where exp(x; q) is the q-exponential given by

exp(x; q) = ∑_{n≥0} \frac{x^{n}}{[n]!_{q}}

using the notation

\begin{aligned}[n]!_{q} & = [n]_{q}[n − 1]_{q} ⋯ [2]_{q}[1]_{q} = \frac{(q; q)_{n}}{(1 − q)^{n}} \\ [n]_{q} & = \frac{1 − q^{n}}{1 − q} \\ (x; q)_{n} & = (1 − x)(1 − qx)(1 − q^{2}x) ⋯ (1 − q^{n−1}x) \end{aligned}

For this reason, we call

W (t, q) = ∑_{w∈W} t^{des(w)}q^{l(w)}

the q-Eulerian distribution of the Coxeter system (W, S), or the q-Eulerian distribution of W by abuse of notation. (We caution the reader that this is not the same notion as the q-Eulerian polynomial considered in [Br] for W = B_{n}, D_{n}). Analogous generating functions to equation (1) for the infinite families of finite Coxeter groups W = B_{n}(= C_{n}), D_{n} were computed in [Re1,Re2].

Note that in the case of an infinite Coxeter group W, the Eulerian distribution ∑_{w∈W} t^{des(w)} does not make sense as a formal power series in t, since there are only finitely many values {0, 1, 2, …, |S| − 1} of des(w) and hence infinitely many group elements w with the same value of des(w). On the other hand, the length distribution

W (q) = ∑_{w∈W} q^{l(w)}

does make sense in ℤ[[q]], and is known to be a computable rational function in q (see equation (6)). The formula for W (t, q) (equation (2)), which essentially comes from inclusion-exclusion, shows that W (t, q) is a computable polynomial in t having coefficients given by rational functions in q. Both this expression for W (t, q) and this corollary are known as folklore within the subject of Coxeter groups, but are hard to find written down.

For some of the classical infinite families of finite and affine Coxeter groups, an encoding trick can be used to produce a generating function encompassing the q-Eulerian distributions of the entire family of groups as in equation (1). We derive a general result (Theorem 4) along these lines, and use it to recover known generating functions for the classical Weyl groups of types A_{n}(= S_{n+1}), B_{n}(= C_{n}), D_{n} (see [St,Re1,Re2]) and derive new results for the infinite families Ã_{n}, B̃_{n}, C̃_{n}, D̃_{n} of affine Weyl groups. For example, we show for the affine Weyl groups S̃_{n}(= Ã_{n−1}) associated to the symmetric groups S_{n} that

∑_{n≥1} \frac{x^{n}}{1 − q^{n}} S̃_{n}(t, q) = [\frac{x\frac{∂}{∂x} log(exp(x; q))}{1 − t exp(x; q)}]_{x↦x \frac{1−t}{1−q}} .

Theorem 4 explains why the factor

1 − t exp(x; q)

naturally appears in the denominator in all of these generating functions.

The paper is structured as follows. Section II collects folklore, known results, and straightforward extensions concerning the computation of the q-Eulerian polynomial W (t, q) of a general Coxeter system (W, S). In Section III, we apply this to compute a generating function analogous to equation (1) for a general class of infinite families of Coxeter groups (Theorem 4). Section IV then specializes this to produce explicit generating functions for all of the infinite families of finite and affine Weyl groups (Theorems 5,6,7,8).

II. How to calculate W (t, q).

We recall here some facts about Coxeter systems (W, S) and refer the reader to [Hu] for proofs and definitions which have been omitted. Given w ∈ W, let its descent set Des(w) be defined by

Des(w) = {s ∈ S : l(ws) < l(w)}

For any subset J ⊆ S, the parabolic subgroup W_{J} is the subgroup generated by J. The set

W^{J} = {w ∈ W : Des(w) ⊆ S − J}

form a set of coset representatives for W/W_{J}, and furthermore when w ∈ W is written uniquely in the form w = u · v where u ∈ W^{J}, v ∈ W_{J}, then we have l(u) + l(v) = l(w). As a consequence,

W_{J}(q) (  ∑_{w∈W:Des(w)⊆S−J} q^{l(w)}) = W (q)

∑_{w∈W:Des(w)⊆S−J} q^{l(w)} = \frac{W (q)}{W_{J}(q)}

where recall that we are using the notation

W (q) = ∑_{w∈W} q^{l(w)}.

We will consider not only subsets S ⊆ T, but also multisets T on the ground set S, which we think of as functions T : S → ℕ specifying a multiplicity T (s) for each element of s in S. For any such function T in S^{ℕ}, let T̂ denote its support, i.e. the subset T̂ ⊆ S defined by

T̂ = {s ∈ S : T (s) > 0}.

Also denote by |T | the cardinality ∑_{s∈S} T (s) of the multiset or function.

Theorem 1. For any Coxeter system (W, S) we have

(2)  W (t, q) = ∑_{T ⊆S} t^{|T |}(1 − t)^{|S−T |} \frac{W (q)}{W_{S−T} (q)}

(3)  \frac{W (t, q)}{(1 − t)^{|S|}} = ∑_{T ∈S^{ℕ}} t^{|T |} \frac{W (q)}{W_{S−T̂}(q)}

Proof. We prove equation (2), from which (3) follows easily. Starting with the right-hand side of (2), one has

∑_{T ⊆S} t^{|T |}(1 − t)^{|S−T |} \frac{W (q)}{W_{S−T} (q)}

= ∑_{T ⊆S} t^{|T |}(1 − t)^{|S−T |} ∑_{w∈W:Des(w)⊆T} q^{l(w)}

= ∑_{w∈W} q^{l(w)} ∑_{Des(w)⊆T ⊆S} t^{|T |}(1 − t)^{|S−T |}

= ∑_{w∈W} q^{l(w)}t^{des(w)} ∑_{∅⊆T′⊆S−Des(w)} t^{|T′|}(1 − t)^{|S−Des(w)−T′|}

= ∑_{w∈W} q^{l(w)}t^{des(w)}(t + (1 − t))^{|S−Des(w)|}

= ∑_{w∈W} q^{l(w)}t^{des(w)}

= W (t, q)■

Remarks. The specialization of equation (2) to q = 1 appears as [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 [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 ∈ 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 additive under every parabolic coset decomposition in the following sense: for all J ⊆ S, when w ∈ W is written uniquely as w = u · v with u ∈ W^{J}, v ∈ W_{J}, we have n(w) = n(u) + n(v). The following theorem is then proven in exactly the same fashion as Theorem 1:

Theorem 1′. Let (W, S) be a Coxeter system, and n_{1}(w), n_{2}(w), … a series of additive statistics. Then using the notations

q^{n(w)} = ∏_{i} q_{i}^{n_{i}(w)}
t^{T} = ∏_{s∈T} t_{s}
(1 − t)^{T} = ∏_{s∈T}(1 − t_{s})
W (q) = ∑_{w∈W} q^{n(w)}
W (t, q) = ∑_{w∈W} t^{Des(w)}q^{n(w)}

we have

(4)  W (t, q) = ∑_{subset T ⊆S} t^{T} (1 − t)^{S−T} \frac{W (q)}{W_{S−T} (q)}

(5)  \frac{W (t, q)}{(1 − t)^{S}} = ∑_{T ∈S^{ℕ}} t^{T} \frac{W (q)}{W_{S−T̂}(q)}■

In light of this theorem, it is useful to know a classification of the additive statistics on W:

Proposition 2. Let (W, S) be a Coxeter system, and let n : W → ℤ be an additive statistic in the above sense. Then

1. The statistic n is completely determined by its values on S via the formula

n(w) = ∑_{j=1}^{l(w)} n(s_{i_{j}})

for any reduced decomposition w = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{l(w)}}.

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. [Hu, Exercise §5.3]) to coincide with the connected components of nodes in the subgraph induced by the odd-labelled edges of the Coxeter diagram.

As a consequence, there is a universal tuple of additive statistics n_{1}, n_{2}, … whose multivariate distribution specializes to that of any other additive statistics, defined by setting n_{i}|_{S} to be the characteristic function of the i^{th} W -conjugacy class restricted to S.

Proof. If n is additive, then the decomposition 1 = 1 · 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 ∈ Des(w), and then w = ws · s is the unique decomposition in W^{{s}} · 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 ⋯ = s′ s s′ ⋯

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) [Hu, §8.1] shows that any such function on S will extend (by the above formula) to a well-defined additive function on W. ■

Recall [Hu, §1.11, §5.12] the fact that W (q) is a rational function in q, which may be computed using the recursion

(6)  W (q) = f(q) (∑_{J⊊S} \frac{(−1)^{|J|}}{W_{J}(q)} )^{−1}

where

f(q) = \{ \begin{matrix}(−1)^{|S|+1} & if W is infinite \\ q^{l(w_{0})} + (−1)^{|S|+1} & if W is finite \end{matrix}

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) ∈ ℤ(q)[t]). More generally, the q-analogue of recursion (6) in which q is replaced by q and l(w) by a(w) follows from the same proof as (6). Therefore W (q) ∈ ℤ(q) for any additive statistics a_{1}(w), a_{2}(w), …, and from equation (4) we conclude that W (t, q) ∈ ℤ(q)[t].

Before leaving this folklore section, we note a happy occurrence when the Coxeter diagram for W is linear, i.e. when it has no nodes of degree greater than or equal to 3. In this situation and with q = 1, Stembridge [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:

Theorem 3. Let (W, S) be a Coxeter system with linear Coxeter diagram, and label the nodes 1, 2, …, n in linear order. Then

W (t, q) = W (q) det[a_{ij}]_{0≤i,j≤n}

where

a_{ij} = \{ \begin{matrix}0 & i − j > 1 \\ t_{i} − 1 & i − j = 1 \\ \frac{t_{i}}{W_{[i+1,j]}(q)} & i ≤ j \end{matrix}

and by convention t_{0} = 1, and W_{[i+1,i]} is the trivial group with 1 element.■

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 signed permutation). It is not hard to check (see e.g. [Re1, Lemma 3.1]) that if we let q^{n(w)} = a^{n(w)}q^{l(w)}, then

B_{n}(q) = (−aq; q)_{n}[n]!_{q}

and hence the above determinant is very explicit. For example when n = 2,

B_{2}(t, q) = (−aq; q)_{2} [2]!_{q} det [ \begin{matrix}1 & \frac{1}{[2]!_{q}} & \frac{1}{(−aq;q)_{2}[2]!_{q}} \\ t_{1} − 1 & t_{1} & \frac{t_{1}}{(−aq;q)_{1}[1]!_{q}} \\ 0 & t_{2} − 1 & t_{2} \end{matrix} ]
= 1 + qt_{1} + aq^{2}t_{1} + aq^{3}t_{1} + aqt_{2} + aq^{2}t_{2} + a^{2}q^{3}t_{2} + a^{2}q^{4}t_{1}t_{2}.

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 ∈ 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 ∈ ℕ 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}, …, 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

∑_{n≥0} \frac{x^{n}}{W^{(n)}(q)}W^{(n)}(t, q)

For a subset J ⊆ S − v, let (W_{J}^{(n)}, S_{J}^{(n)}) be the Coxeter system corresponding to the parabolic subgroup generated by J ∪ {s_{0}, …, s_{n}}. Also define for J ⊆ S − v and a, b ∈ ℕ 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 ∪ ({s_{0}, …, s_{n}} − s_{a}). Let

\begin{aligned}exp_{W_{J}} (x; q) & = ∑_{n≥0} \frac{x^{n}}{W_{J}^{(n)}(q)} \\ dex_{W_{J}} (x; q) & = ∑_{a,b≥0} \frac{x^{a+b}}{W_{J}^{(a,b)}(q)} \end{aligned}

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 double exponential) or the derivative of such a q-analogue.

Theorem 4.

∑_{n≥0} \frac{x^{n}}{W^{(n)}(q)}W^{(n)}(t, q) =

[ ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|} (exp_{W_{S−v−J}} (x; q) + \frac{t dex_{W_{S−v−J}} (x; q)}{1 − t exp(x; q)})]_{x↦x(1−t)}

Proof. From equation (2) we have

W^{(n)}(t, q) = ∑_{T ⊆S^{(n)}} t^{|T |}(1 − t)^{|S^{(n)}−T |} \frac{W^{(n)}(q)}{W_{S^{(n)}−T}^{(n)} (q)}

so that

\frac{W^{(n)}(t, q)}{W^{(n)}(q) (1 − t)^{n}}

= ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|} ∑_{K⊆\{s_{0},... ,s_{n}\}} \frac{t^{|K|}}{(1 − t)^{|K|}} \frac{1}{W_{S^{(n)}−J−K}^{(n)}(q)}

= ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|} ∑_{K∈ℕ^{\{s_{0},... ,sn\}}} t^{|K|} \frac{1}{W_{S^{(n)}−J−K̂}^{(n)}(q)}

= ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|} (\frac{1}{W_{S−v−J}^{(n)}(q)} + ∑_{k≥1} t^{k} ∑_{{K∈ℕ^{\{s_{0},... ,sn\}} \atop |K|=k}} \frac{1}{W_{S^{(n)}−J−K̂}^{(n)}(q)})

At this stage, we use an encoding for the functions K : {s_{0}, …, s_{n}} → ℕ having |K| = k. Let ω_{i} ∈ ℝ^{n} be the vector e_{1} + e_{2} + … + e_{i}, where e_{i} is the i^{th} standard basis vector, so that ω_{0} = (0, 0, …, 0) and ω_{n} = (1, 1, …, 1). Given K : {s_{0}, …, s_{n}} → ℕ, encode it as the vector c(K) = ∑_{i=0}^{n} K(s_{i}) ω_{i} ∈ ℝ^{n}. Note that once we have fixed the cardinality |K| = k ≥ 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}, …, 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−K̂} is then isomorphic to

W_{S−v−J}^{(a,b)} × S_{a_{1}} × ⋯ × S_{a_{k−1}}.

Therefore we may continue the calculation

\frac{W^{(n)}(t, q)}{W^{(n)}(q) (1 − t)^{n}} = ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|}×

(\frac{1}{W_{S−v−J}^{(n)}(q)} + ∑_{k≥1} t^{k} ∑_{(a_{0},... ,a_{k})∈ℕ^{k+1} ; ∑ a_{i}=n} \frac{1}{W_{S−v−J}^{(a,b)}(q) [a_{1}]!_{q} ⋯ [a_{k−1}]!_{q}} )

∑_{n≥0} \frac{W^{(n)}(t, q)}{W^{(n)}(q)} \frac{x^{n}}{(1 − t)^{n}} = ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|}×

(∑_{n≥0} \frac{x^{n}}{W_{S−v−J}^{(n)}(q)} + ∑_{k≥1} t^{k} ∑_{n≥0} ∑_{(a_{0},... ,a_{k})∈ℕ^{k+1} ; ∑ a_{i}=n} \frac{x^{a_{0}+a_{k}}}{W_{S−v−J}^{(a_{0},a_{k})}(q)} \frac{x^{a_{1}}}{[a_{1}]!_{q}} ⋯ \frac{x^{a_{k−1}}}{[a_{k−1}]!_{q}} )

= ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|}×

(exp_{W_{S−v−J}} (x; q) + ∑_{a_{0},a_{k}≥0} \frac{x^{a_{0}+a_{k}}}{W_{S−v−J}^{(a,b)}(q)} ∑_{k≥1} t^{k}(exp(x; q))^{k})

= ∑_{J⊆S−v} t^{|J|}(1 − t)^{|S−J|} (exp_{W_{S−v−J}} (x; q) + dex_{W_{S−v−J}} (x; q)\frac{t}{1 − t exp(x; q)})

The theorem now follows upon replacing x by x(1 − t).■

Remarks.

1. The crucial encoding of functions K : {s_{0}, …, s_{n}} → ℕ used in the middle of the preceding proof is a translation and generalization of the “direct encoding” used in [GG, §1] for type A_{n}.

2. There is an obvious q-analogue of Theorem 3 involving additive statistics on (W, S), with the same proof.

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 Weyl group of rank n, then there is an associated multiset of numbers e_{1}, e_{2}, …, e_{n} called the exponents of W, satisfying

(7)  W (q) = ∏_{i=1}^{n}[e_{i} + 1]_{q}

(8)  W̃ (q) = ∏_{i=1}^{n} \frac{[e_{i} + 1]_{q}}{1 − q^{e_{i}}}

where W̃ is the affine Weyl group associated to W. The first formula is a theorem of Chevalley [Hu, §3.15], the second a theorem of Bott [Hu, §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 [BB, BE , EE, ER].

We first consider an infinite family of Coxeter systems with linear diagrams. Let W_{n}^{r,s} 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 except for the leftmost edge labelled r and the rightmost edge labelled s. Let W_{n}^{r} be the family defined by W_{n}^{r} = W_{n}^{r,3} The next result uses Theorem 4 to compute a generating function for W_{n}^{r,s}(t, q). Note that W_{n}^{r,s} 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 C̃_{n}, as well as some hyperbolic Coxeter groups (see [Hu, §2.4, 2.5, 6.9]).

Before stating the theorem, we establish some more notation. Let

\begin{aligned}exp_{W^{r}}(x; q) & = ∑_{n≥0} \frac{x^{n}}{W_{n}^{r}(q)} \\ exp_{W^{r,s}}(x; q) & = ∑_{n≥0} \frac{x^{n}}{W_{n}^{r,s}(q)} \end{aligned}

where by convention we define W_{0}^{r,s} = W_{0}^{r} to be the trivial group with 1 element, W_{1}^{r,s} = W_{1}^{r} is the unique Coxeter system of rank 1, and W_{2}^{r,s} = W_{2}^{r} = I_{2}(r) is the rank 2 (dihedral) Coxeter system of order 2r.

Theorem 5.

(9) ∑_{n≥0} \frac{x^{n}}{W_{n}^{r,s}(q)}W_{n}^{r,s}(t, q) = exp_{W^{r,s}}(x(1 − t); q)

+ \frac{t x (1 − t) exp_{W^{r}}(x(1 − t); q) exp_{W^{s}}(x(1 − t); q)}{1 − t exp(x(1 − t); q)}

(10) ∑_{n≥0} \frac{x^{n}}{W_{n}^{r}(q)}W_{n}^{r}(t, q) = \frac{(1 − t) exp_{W^{r}}(x(1 − t); q)}{1 − t exp(x(1 − t); q)}

Proof. Equation (10) follows from equation (9) by setting s = 3 and noting that

exp_{W^{r,3}}(x; q) = exp_{W^{r}}(x; q)
exp_{W^{3}}(x; q) = \frac{exp(x; q) − 1}{x}.

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

W^{(n)} = W_{n+3}^{r,s}
exp_{W_{s_{1},s_{3}}} (x; q) = x^{−3} (exp_{W^{r,s}}(x; q) − 1 − \frac{x}{[2]_{q}} − \frac{x^{2}}{[2]_{q}[r]_{q}} )
exp_{W_{s_{1}}} (x; q) = x^{−2} (exp_{W^{r}}(x; q) − 1 − \frac{x}{[2]_{q}} )
exp_{W_{s_{3}}} (x; q) = x^{−2} (exp_{W^{s}}(x; q) − 1 − \frac{x}{[2]_{q}} )
exp_{W_{∅}}(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_{∅}}(x; q) = x^{−2}(exp(x; q) − 1)^{2}

and using these facts, equation (9) follows from Theorem 4 with a little algebra. ■

We now specialize Theorem 5 to obtain generating functions for the types A_{n−1}(= S_{n}), B_{n}(= C_{n}), and C̃_{n}.

If r = 3 then W_{n}^{r} coincides with the finite Weyl group A_{n}(= S_{n+1}) which has exponents 1, …, 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) = ∑_{n≥0} \frac{(x(1 − q))^{n}}{(q; q)_{n}} = (x(1 − q); q)_{∞}^{−1}

where the last equality is by the q-binomial theorem [GR, Appendix II.3]:

∑_{n≥0} \frac{(z; q)_{n}}{(q; q)_{n}} x^{n} = \frac{(zx; q)_{∞}}{(x; q)_{∞}} .

If r = 4 then W_{n}^{r} coincides with the finite Weyl group B_{n} or C_{n} which has exponents 1, 3, …, 2n − 1. In this case equation (10) is equivalent to [Re1, §3] 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) = ∑_{n≥0} \frac{(x(1 − q))^{n}}{(q^{2}; q^{2})_{n}} = (x(1 − q); q^{2})_{∞}^{−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}(q) = ∑_{w∈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 [Re1, §3, 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) = ∑_{n≥0} \frac{x^{n}}{(−aq; q)_{n}[n]!_{q}}

has a nice infinite product expression.

If r = s = 4 then W_{n+1}^{r,s} coincides with the affine Weyl group C̃_{n} for n ≥ 2, so equation (9) says

1 + \frac{x(1 + tq)}{[2]_{q}} + \frac{x^{2}C_{2}(t, q)}{[2]_{q}[4]_{q}} + x ∑_{n≥2} \frac{x^{n}}{C̃_{n}(q)}C̃_{n}(t, q)

= exp_{W^{4,4}}(x(1 − t); q) + \frac{t x (1 − t) [exp_{W^{4}}(x(1 − t); q)]^{2}}{1 − t exp(x(1 − t); q)} .

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, …, 2n − 1, by (8) we have

(11)  W_{n+1}^{4,4} = C̃_{n}(q) = \frac{(q^{2}; q^{2})_{n}}{(1 − q)^{n}(q; q^{2})_{n}}

for n ≥ 2 and hence

exp_{W^{4,4}}(x; q) = 1 + \frac{x}{[2]_{q}} + \frac{x^{2}}{[2]_{q}[4]_{q}} + x ∑_{n≥2} \frac{(q; q^{2})_{n}}{(q^{2}; q^{2})_{n}} (x(1 − q))^{n}

= 1 + \frac{x}{[2]_{q}} + \frac{x^{2}}{[2]_{q}[4]_{q}} + x (\frac{(xq(1 − q); q^{2})_{∞}}{(x(1 − q); q^{2})_{∞}} − \frac{x(1 − q)}{1 + q} − 1)

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} = C̃_{n}:

C̃_{n}(q) = ∑_{w∈C̃_{n}} a^{n(w)}b^{m(w)}q^{l(w)}

= \frac{(−aq; q)_{n}(−bq; q)_{n}[n]!_{q}}{(abq^{n+1}; q)_{n}} (12)

by using the q-generalization of recursion (6) to show

\frac{1}{C̃_{n}(q)} = ∑_{i=0}^{n} \frac{q^{i^{2}}a^{i}}{B_{i}(a, q)B_{n−i}(b, q)}

= ∑_{i=0}^{n} \frac{q^{i^{2}}a^{i}}{(−aq; q)_{i}[i]!_{q}(−bq; q)_{n−i}[n − i]!_{q}}

and then applying the q-Vandermonde summation formula [GR, Appendix II.6]. The q-refinement of Theorem 5 with r = s = 4 then gives a very explicit generating function generalization enumerating C̃_{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

exp_{W^{4,4}}(x; a, b, q) = ∑_{n≥0} \frac{(abq^{n+1}; q)_{n}}{(−bq; q)_{n}(−aq; q)_{n}[n]!_{q}} x^{n}

= _{4}ϕ_{3} ( \begin{matrix}\sqrt{abq} & −\sqrt{abq} & \sqrt{ab}q & −\sqrt{ab}q \\ −aq & −bq & abq \end{matrix} | x(1 − q); q)

where the last equation is basic hypergeometric series notation (see e.g. [GR]).

We next deal with the affine symmetric groups. Let Ã_{n−1} = S̃_{n} be the affine Weyl group corresponding to the Weyl group A_{n−1} = S_{n}, so S̃_{n} has as its Coxeter diagram a cycle with n vertices and label 3 on every edge. Let

expS̃(x : q) = ∑_{n≥1} \frac{x^{n}}{S̃_{n}(q)}.

We now prove a formula claimed in the Introduction:

Theorem 6.

∑_{n≥1} \frac{x^{n}}{1 − q^{n}} S̃_{n}(t, q) = [\frac{x\frac{∂}{∂x} log(exp(x; q))}{1 − t exp(x; q)}]_{x↦x \frac{1−t}{1−q}} .

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

the electronic journal of combinatorics 2 (1995), #R25 14

v = s_{3} and B_{1} = {s_{1}}, B_{2} = {s_{2}}. One can then check that

W^{(n)} = S̃_{n+3}
exp_{W_{s_{1},s_{2}}} (x; q) = x^{−3} (expS̃(x; q) − x − \frac{x^{2}(1 − q)}{[2]_{q}})
exp_{W_{s_{1}}} (x; q) = exp_{W_{s_{2}}} (x; q) = x^{−3} (exp(x; q)1 − x − \frac{x^{2}}{[2]_{q}} )
exp_{W_{∅}}(x; q) = x^{−2}(exp(x; q) − 1 − x)
dex_{W_{s_{1},s_{2}}} (x; q) = −2x^{−3} (exp(x; q) − 1 − x − \frac{x^{2}}{[2]_{q}} )

+ x^{−2} (\frac{∂}{∂x} exp(x; q) − 1 − \frac{2x}{[2]_{q}} )

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_{∅}}(x; q) = x^{−2}(exp(x; q) − 1)^{2}

and using these facts one can simplify Theorem 4 in this case to

∑_{n≥1} \frac{x^{n}}{S̃_{n}(q)}S̃_{n}(t, q) = [expS̃(x; q) + \frac{t x \frac{∂}{∂x} exp(x; q)}{1 − t exp(x; q)}]_{x↦x (1−t)} .

To rewrite this more explicitly, we note that the exponents of S_{n} = A_{n−1} are 1, 2, …, n − 1, so that equation (8) gives

S̃_{n}(q) = \frac{[n]!_{q}}{(q; q)_{n−1}} = \frac{1 − q^{n}}{(1 − q)^{n}} .

Therefore

expS̃(x; q) = ∑_{n≥1} \frac{x^{n}(1 − q)^{n}}{1 − q^{n}}

= ∑_{n≥1} ∑_{m≥0}(x(1 − q))^{n}q^{nm}

= ∑_{m≥0} ∑_{n≥1}(x(1 − q)q^{m})^{n}

= ∑_{m≥0} \frac{x(1 − q) q^{m}}{1 − x(1 − q) q^{m}}

= ∑_{m≥0} x\frac{∂}{∂x} log[(1 − x(1 − q) q^{m})^{−1}]

= x\frac{∂}{∂x} log[(x(1 − q); q)_{∞}^{−1}]

= x\frac{∂}{∂x} log(exp(x; q))

Substituting this into the last equation and replacing x by \frac{x}{1−q} gives

∑_{n≥1} \frac{x^{n}}{1 − q^{n}} S̃_{n}(t, q) = [x\frac{∂}{∂x} log(exp(x; q)) + \frac{t x \frac{∂}{∂x} exp(x; q)}{1 − t exp(x; q)}]_{x↦x \frac{1−t}{1−q}}

which is equivalent to the theorem by a little algebra.■

Next we move on to a common generalization of the Weyl groups D_{n} and the affine Weyl groups B̃_{n}. Let D_{n}^{r} 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_{n}^{3} = D_{n} and D_{n}^{4} = B̃_{n} (see [Hu, §2.4, 2.5]). We adopt the notation

exp_{D^{r}}(x; q) = ∑_{n≥2} \frac{x^{n}}{D_{n}^{r}(q)}

exp_{D}(x; q) = ∑_{n≥2} \frac{x^{n}}{D_{n}(q)}

where by convention we define D_{2}^{r} = A_{1} ⊕ A_{1} and D_{3}^{r} = A_{3}. Remark: The notation exp_{D}(x; q) is slightly different from the notation exp_{D}(u) used in [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) = ∑_{n≥0} \frac{u^{n}}{(−q; q)_{n−1}[n]!_{q}}

should actually read

exp_{D}(u) = 2 + ∑_{n≥1} \frac{u^{n}}{(−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}.

Theorem 7.

∑_{n≥4} \frac{x^{n}}{D_{n}^{r}(q)}D_{n}^{r}(t, q) =

∑_{n≥4}⟨x^{n}⟩ [exp_{D^{r}}(x; q) + \frac{t x exp_{W^{r}}(x; q)}{1 − t exp(x; q)} (2 − \frac{tx}{1 − t} + exp_{D}(x; q))]_{x↦x(1−t)} · x^{n}

(13)

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

W^{(n)} = D_{n+4}^{r}
exp_{W_{s_{1},s_{2},s_{3}}} (x; q) = x^{−4} (exp_{D^{r}}(x; q) − \frac{x^{2}}{([2]_{q})^{2}} − \frac{x^{3}}{[3]!_{q}} )
exp_{W_{s_{1},s_{2}}} (x; q) = x^{−3} (exp_{D}(x; q) − \frac{x^{2}}{([2]_{q})^{2}} )
exp_{W_{s_{1},s_{3}}} (x; q) = exp_{W_{s_{2},s_{3}}} (x; q) = x^{−3} (exp_{W^{r}}(x; q) − 1 − \frac{x}{[2]!_{q}} − \frac{x^{2}}{[2]_{q}[r]_{q}} )
exp_{W_{s_{1}}} (x; q) = exp_{W_{s_{2}}} (x; q) = x^{−3} (exp(x; q) − 1 − x − \frac{x^{2}}{[2]!_{q}} )
exp_{W_{s_{3}}} (x; q) = x^{−2} (exp_{W^{r}}(x; q) − 1 − \frac{x}{[2]!_{q}} )
exp_{W_{∅}}(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_{∅}}(x; q) = x^{−2}(exp(x; q) − 1)^{2}

and using these the result follows from Theorem 4. ■

We now specialize Theorem 7 to r = 3, 4. If r = 3, then D_{n}^{r} = 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 ≥ 4). Therefore we obtain

2tx + ∑_{n≥2} \frac{x^{n}}{D_{n}(q)}D_{n}(t, q) =

\frac{(1 − t) exp_{D}(x(1 − t); q) + t (2 − tx) (exp(x(1 − t); q) − 1)}{1 − t exp(x(1 − t); q)}.

which one can easily check agrees with [Re2, Corollary 4.5]. Note that since D_{n} has exponents 1, 3, …, 2n − 3, n − 1 by equation (7) we have

exp_{D}(x; q) = ∑_{n≥2} \frac{x^{n}}{(−q; q)_{n−1}[n]!_{q}}

= ∑_{n≥2} \frac{x^{n}(1 − q)^{n}(1 + q^{n})}{(q^{2}; q^{2})_{n}}

= ∑_{n≥2} (\frac{(x(1 − q))^{n}}{(q^{2}; q^{2})_{n}} + \frac{(xq(1 − q))^{n}}{(q^{2}; q^{2})_{n}} )

= (x(1 − q); q^{2})_{∞}^{−1} + (xq(1 − q); q^{2})_{∞}^{−1} − 2 − x

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_{n}^{r} = B̃_{n−1}, so equation (13) gives a closed form for the generating function

∑_{n≥3} \frac{x^{n}}{B̃_{n}(q)}B̃_{n}(t, q).

Since B̃_{n} is the affine Weyl group associated to B_{n}, which has the same exponents as C_{n}, we must have B̃_{n}(q) = C̃_{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 B̃_{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 = B̃_{n}:

B̃_{n}(q) = ∑_{w∈B̃_{n}} a^{n(w)}q^{l(w)}

= \frac{(−aq; q)_{n}(−q; q)_{n−1}[n]!_{q}}{(aq^{n}; q)_{n}} (14)

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 D̃_{n}, whose Coxeter diagram looks like a path with forks at both ends having n nodes total, and all edges labelled 3 (see [Hu, §2.5]). We use the notation

expD̃(x; q) = ∑_{n≥4} \frac{x^{n}}{D̃_{n}(q)}.

Theorem 8.

∑_{n≥4} \frac{x^{n}}{D̃_{n}(q)}D̃_{n}(t, q) = ∑_{n≥4}⟨x^{n}⟩

\{(1 − t) expD̃(x(1 − t); q) + \frac{t}{1 − t exp(x(1 − t); q)}×

[\frac{t^{2}(2 + 2x − tx)^{2} + t(2 + tx)(2 − 4t − 3tx + 2t^{2}x) exp(x(1 − t); q)}{1 − t}

+2(1 − t)(2 − tx) exp_{D}(x(1 − t); q) + (1 − t) exp_{D}(x(1 − t); q)^{2}]} x^{n}

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

W^{(n)} = D̃_{n+4}^{r}
exp_{W_{s_{1},s_{2},s_{3},s_{4}}} (x; q) = x^{−4}expD̃(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} (exp_{D}(x; q) − \frac{x^{2}}{([2]_{q})^{2}} − \frac{x^{3}}{[3]!_{q}} )

exp_{W_{s_{1},s_{2}}} (x; q) = exp_{W_{s_{3},s_{4}}} (x; q) = x^{−3} (exp_{D}(x; q) − \frac{x^{2}}{([2]_{q})^{2}} )
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} (exp(x; q) − 1 − x − \frac{x^{2}}{[2]!_{q}} − \frac{x^{3}}{[3]!_{q}} )

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} (exp(x; q) − 1 − x − \frac{x^{2}}{[2]!_{q}} )

exp_{W_{∅}}(x; q) = x^{−2}(exp(x; q) − 1 − x)
dex_{W_{s_{1},s_{2},s_{3},s_{4}}} (x; q) = x^{−4} exp_{D̃(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_{∅}}(x; q) = x^{−2}(exp(x; q) − 1)^{2}

and using these facts, the result follows from Theorem 4 with a little algebra. ■

As in the previous cases of affine Weyl groups, it is possible to replace expD̃(x) by an expression involving infinite products, if desired. Since D_{n} has exponents 1, 3, 5, …, 2n − 5, 2n − 3, n − 1, by equation (8) we have

D̃_{n}(q) = \frac{(−q; q)_{n−1}[n]!_{q}}{(1 − q)^{n}(q; q^{2})_{n−1}(1 − q^{n−1})}

= \frac{(q^{2}; q^{2})_{n}}{(1 − q)^{n}(q^{−1}; q^{2})_{n}} \frac{(1 − q^{−1})}{(1 + q^{n})(1 − q^{n−1})}

Therefore

expD̃(x; q) = ∑_{n≥4}(x(1 − q))^{n} \frac{(q^{−1}; q^{2})_{n}}{(q^{2}; q^{2})_{n}} \frac{(1 + q^{n})(1 − q^{n−1})}{(1 − q^{−1})}

= ∑_{n≥4}(x(1 − q))^{n} \frac{(q^{−1}; q^{2})_{n}}{(q^{2}; q^{2})_{n}} [(1 + q^{n}) + \frac{1}{q − 1}(1 − q^{2n})]

= \frac{(xq^{−1}(1 − q); q^{2})_{∞}}{(x(1 − q); q^{2})_{∞}} + \frac{(x(1 − q); q^{2})_{∞}}{(xq(1 − q); q^{2})_{∞}}

+ xq^{−1}(1 − q)\frac{(xq(1 − q); q^{2})_{∞}}{(x(1 − q); q^{2})_{∞}} − ∑_{i=0}^{3} c_{i}(q) x^{i}

= \frac{(xq(1 − q); q^{2})_{∞}}{(x(1 − q); q^{2})_{∞}} + \frac{(x(1 − q); q^{2})_{∞}}{(xq(1 − q); q^{2})_{∞}} − ∑_{i=0}^{3} c_{i}(q) x^{i}

for some c_{i}(q) which are rational functions of q.

Acknowledgements. The author would like to thank Anders Björner, 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.

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School of Mathematics, University of Minnesota, Minneapolis, MN 55455