h-vectors of generalized associahedra and non-crossing partitions

Abstract

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