Sweeping the cd-index and the toric \(h\)-vector (Q540054)

Summary: We derive formulas for the cd-index and the toric \(h\)-vector of a convex polytope \(P\) from a sweeping by a hyperplane. These arise from interpreting the corresponding \(S\)-shelling of the dual of \(P\). We describe a partition of the faces of the complete truncation of \(P\) to reflect explicitly the nonnegativity of its cd-index and what its components are counting. One corollary is a quick way to compute the toric \(h\)-vector directly from the cd-index that turns out to be an immediate consequence of formulas of Bayer and Ehrenborg. We also propose an ``extended toric'' \(h\)-vector that fully captures the information in the flag \(h\)-vector.