Linear Inequalities for Flags in Graded Partially Ordered Sets

The closure of the convex cone generated by all flag f-vectors of graded partially ordered sets 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 partially ordered sets. 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 partially ordered sets of rank at most 5.