h-VECTORS OF GENERALIZED ASSOCIAHEDRA AND NONCROSSING PARTITIONS

A case-free proof is given that the entries of the h-vector of the cluster complex ∆(Φ), associated by S. Fomin and A. Zelevinsky to a finite root system Φ, count elements of the lattice L of noncrossing partitions of corresponding type by rank. Similar interpretations for the h-vector of the positive part of ∆(Φ) are provided. The proof utilizes the appearance of the complex ∆(Φ) in the context of the lattice L, in recent work of two of the authors, as well as an explicit shelling of ∆(Φ).