Polygon dissections and some generalizations of cluster complexes

Let W be a Weyl group corresponding to the root system A n − 1 or B n . We define a simplicial complex Δ W m in terms of polygon dissections for such a group and any positive integer m. For m = 1 , Δ W m is isomorphic to the cluster complex corresponding to W, defined in [S. Fomin, A.V. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977–1018]. We enumerate the faces of Δ W m and show that the entries of its h-vector are given by the generalized Narayana numbers N W m ( i ) , defined in [C.A. Athanasiadis, 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 Δ W m is shellable and hence Cohen–Macaulay.