The toric h-vector of a cubical complex in terms of noncrossing partition statistics

This paper introduces a new and simple statistic on noncrossing partitions that expresses each coordinate of the toric $h$-vector of a cubical complex, written in the basis of the Adin $h$-vector entries, as the total weight of all noncrossing partitions. The same model may also be used to obtain a very simple combinatorial interpretation of the contribution of a cubical shelling component to the toric $h$-vector. In this model, a strengthening of the symmetry expressed by the Dehn-Sommerville equations may be derived from the self-duality of the noncrossing partition lattice, exhibited by the involution of Simion and Ullman

Linear inequalities for flags in graded posets

The closure of the convex cone generated by all flag $f$-vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5