28th Cumberland Conference on Combinatorics, Graph Theory & Computing
Gábor Hetyei
University of North Carolina, Charlotte http://math2.uncc.edu/~ghetyei/ 

Abstract 
Cubical $h$vectors, noncrossing partitions, and Catalan numbers
After a brief introduction to $h$vectors of cubical complexes, we express each coordinate of Stanley's toric $h$vector, written in the basis of the Adin $h$vector entries, as the total weight of all noncrossing partitions in a weighted enumeration model. In our model, the symmetry expressed by the DehnSommerville equations is a consequence of the selfduality of the noncrossing partition lattice, exhibited by the involution of Simion and Ullman. By collecting the appropriate terms we also obtain a new, very simple combinatorial interpretation of the contribution of a shelling component to the toric hvector of a cubical complex. 