Posets from Polytopes and Refined Face Enumeration

(Talk by Henri Muehle)

Abstract: The 1-skeleton of a polytopal complex can be oriented by means of an appropriate cost function.  We are particularly interested in those cases, when the orientation yields the poset diagram of a lattice.  One particularly fascinating instance of this construction is the Tamari lattice arising from an orientation of the simple associahedron.
Using this perspective, we explain how two remarkable polynomials, the F- and the H-triangle, introduced by F. Chapoton in the context of root systems can be realized in terms of the Tamari lattice.  We give a simple, combinatorial proof of the F=H-correspondence asserting that the F- and the H-triangle are related by substition of variables.  This correspondence generalizes the well-known correspondence between the f- and the h-vector of a polytope.
This talk is based on joint work with Cesar Ceballos from TU Graz.