Polygon dissections and some generalizations of cluster complexes

Abstract

Let $W$ be a Weyl group corresponding to the root system $A_{n-1}$ or $B_n$. We define a simplicial complex ${\Delta }_W^m$ in terms of polygon dissections for such a group and any positive integer $m$. For $m=1$, ${\Delta }_W^m$ is isomorphic to the cluster complex corresponding to $W$, defined in \cite{FZ}. We enumerate the faces of ${\Delta }_W^m$ and show that the entries of its $h$-vector are given by the generalized Narayana numbers $N_W^m(i)$, defined in \cite{Atha3}. We also prove that for any $m\ge 1$ the complex ${\Delta }_W^m$ is shellable and hence Cohen-Macaulay.

9 pages, 3 figures, the type D case has been removed, some corrections on the proof of Theorem 3.1 have been made. To appear in JCTA