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Séminaire Lotharingien de Combinatoire, B66c (2011), 21 pp.

# Christos A. Athanasiadis and Christina Savvidou

# The Local *h*-Vector of the Cluster Subdivision of a Simplex

**Abstract.**
The cluster complex *\Delta*(*\Phi*) is an abstract simplicial complex,
introduced by Fomin and Zelevinsky for a finite root system *\Phi*.
The positive part of *\Delta*(*\Phi*) naturally defines a simplicial
subdivision of the simplex on the vertex set of simple roots of *\Phi*.
The local *h*-vector of this subdivision, in the sense of Stanley, is
computed and the corresponding *\gamma*-vector is shown to be
nonnegative. Combinatorial interpretations to the entries of the local
*h*-vector and the corresponding *\gamma*-vector are provided for the
classical root systems, in terms of noncrossing partitions of types *A*
and *B*. An analogous result is given for the barycentric subdivision
of a simplex.

Received: October 12, 2011.
Accepted: February 24, 2012.
Final Version: March 15, 2012.

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