Polygon dissections and some generalizations of cluster complexes

摘要

Let W be a Weyl group corresponding to the root system A(n-1) or B-n. We define a simplicial complex Delta(m)(W) in terms of polygon dissections for such a group and any positive integer m. For m = 1, Delta(m)(W) is isomorphic to the cluster complex corresponding to W, defined in [S. Fomin, AX Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977-1018]. We enumerate the faces of Delta(m)(W) and show that the entries of its h-vector are given by the generalized Narayana. numbers N-W(m) (i), defined in W [C.A. Athamasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005) 179-196]. We also prove that for any m >= 1 the complex Delta(m)(W) is shellable and hence Cohen-Macaulay. (c) 2005 Elsevier Inc. All rights reserved.