The distribution of descents and length in a Coxeter group

Victor Reiner

The electronic journal of combinatorics, Tome 2 (1995)

Résumé

We give a method for computing the $q$-Eulerian distribution $$ W(t,q)=\sum_{w \in W} t^{{\rm des}(w)} q^{l(w)} $$ 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 ${\rm des}(w)$ is the number of simple reflections $s \in S$ 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.

Détail

DOI : 10.37236/1219

Classification : 20F55, 20F05, 05A15
Mots-clés : Eulerian distributions, rational functions, Coxeter systems, length functions, simple reflections, generating functions, affine Weyl groups