The toric h-vectors of partially ordered sets

An explicit formula for the toric h-vector of an Eulerian poset in terms of the cd-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the ag h-vector. For this, another proof base d on Fine's algorithm and lattice-path counts is given. As a consequence, i t is shown that the Kalai relation on dual posets, g (n/2)(P) = g(n/2)(P*), is the only equation relating the h-vectors of posets and their duals. A r esult on the h-vectors of oriented matroids is given. A simple formula for the cd-index in terms of the ag h-vector is derived.