h-vectors of generalized associahedra and non-crossing partitions

A case-free proof is given that the entries of the $h$-vector of the cluster complex $\Delta (\Phi)$, associated by S. Fomin and A. Zelevinsky to a finite root system $\Phi$, 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 $\Delta (\Phi)$ are provided. The proof utilizes the appearance of the complex $\Delta (\Phi)$ in the context of the lattice $\nc$, in recent work of two of the authors, as well as an explicit shelling of $\Delta (\Phi)$.