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h-vectors of generalized associahedra and non-crossing partitions

Christos A. Athanasiadis·Thomas Brady·Jon McCammond·Colum Watt
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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 $Δ(Φ)$.

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Christos A. Athanasiadis, Thomas Brady, Jon McCammond, Colum Watt. 2006-02-14. h-vectors of generalized associahedra and non-crossing partitions. https://arxiv.org/abs/math/0602293

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