%% 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. 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