Geodesic
The Local h-Vector of the Cluster Subdivision of a Simplex
Séminaire lotharingien de combinatoire, Tome 66 (2011-2012)

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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. 

@article{SLC_2011-2012_66_a2,
     author = {Christos A. Athanasiadis and Christina Savvidou},
     title = {The {Local} {h-Vector} of the {Cluster} {Subdivision} of a {Simplex}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {66},
     year = {2011-2012},
     url = {http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a2/}
}
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Christos A. Athanasiadis; Christina Savvidou. The Local h-Vector of the Cluster Subdivision of a Simplex. Séminaire lotharingien de combinatoire, Tome 66 (2011-2012). http://geodesic.mathdoc.fr/item/SLC_2011-2012_66_a2/