Combinatorics Seminar: Ehrhart Quasipolynomials of Coxeter Permutahedra

Abstract: The Ehrhart polynomial counts lattice points in a dilated lattice polytope. The Ehrhart polynomials of permutahedra of types A, B, C, and D have been calculated by Federico Ardila, Federico Castillo, and Michael Henley (2015). However, when a type B permutahedron is shifted so that its center is the origin, it becomes a half-integral polytope, and its Ehrhart quasipolynomial was previously unknown. The same is true of odd-dimension type A permutahedra. We use signed graphs that arise from the generating vectors of each permutahedron to determine which sets of vectors are linearly independent and thus which form parallelepipeds that are a part of a zonotopal decomposition, as well as which of these parallelepipeds stays on the lattice when the permutahedron is shifted. This yields new approaches/formulas for Ehrhart quasipolynomials for these rational permutahedra.

Host: Laura Escobar