On a refinement of the generalized Catalan numbers for Weyl groups

Abstract

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check {Q}$ and Coxeter number $h$, spanning a Euclidean space $V$, and let $m$ be a positive integer. It is known that the set of regions into which the fundamental chamber of $W$ is dissected by the hyperplanes in $V$ of the form $(\alpha , x) = k$ for $\alpha \in \Phi$ and $k = 1, 2,\dots ,m$ is equinumerous to the set of orbits of the action of $W$ on the quotient $\check {Q} / (mh+1) \check {Q}$. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of $\Phi$, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case $m=1$ and $\Phi = A_{n-1}$.