The toric h-vectors of partially ordered sets

An explicit formula for the toric /i-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 flag 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*), is the only equation relating the h-vectors of posets and their duals. A result on the h-vectors of oriented matroids is given. A simple formula for the cd-index in terms of the flag h-vector is derived.