Department of Mathematics

Generalizing a conjecture by De Loera et al., we conjecture that all the integral generalized permutohedra have positive Ehrhart coefficients. Berline-Vergne constructe a valuation that assign values to faces of polytopes, which provides a way to write Ehrhart coefficients of a polytope as positive sums of these values. Based on empirical results, we conjecture Berline-Vergne's valuation is always positive on regular permutohedra, which implies our first conjecture. This article proves that our conjecture on Berline-Vergne's valuation is true for dimension up to $6$, and is true if we restrict to faces of codimension up to $3.$ We also give two equivalent statements to this conjecture in terms of mixed valuations and Todd class, respectively. In addition to investigating the positivity conjectures, we study the Berline-Vergne's valuation, and show that it is the unique construction for McMullen's formula (used to describe number of lattice points in a polytope) under certain symmetry constraints.