The toric ℎ-vectors of partially ordered sets

Abstract

An explicit formula for the toric h h -vector of an Eulerian poset in terms of the c d \mathbf {cd} -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag h h -vector. For this, another proof based on Fine’s algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, g n / 2 ( P ) = g n / 2 ( P ∗ ) g_{n/2}(P)=g_{n/2}(P^*) , is the only equation relating the h h -vectors of posets and their duals. A result on the h h -vectors of oriented matroids is given. A simple formula for the c d \mathbf {cd} -index in terms of the flag h h -vector is derived.