Krishna Menon: Ehrhart positivity of cross-polytopes

Abstract: The Ehrhart polynomial of a lattice polytope counts lattice points in its dilates. While not true for polytopes in general, there are interesting classes of polytopes whose Ehrhart polynomials have positive coefficients. Cross-polytopes are such a class. Previous proofs of Ehrhart positivity for these polytopes were either inductive or used properties of roots of their Ehrhart polynomials. We present a combinatorial proof by showing that these Ehrhart polynomials are generating functions for certain coloured permutations. This is based on joint work with Emil Verkama.