Combinatorial Theory

A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, which is a rational function with the numerator called the \(h^\ast\)-polynomial. We compute the \(h^\ast\)-polynomials of an arbitrary positroid polytope by a family of shelling orders of it. We also compute the \(h^\ast\)-polynomial of any positroid polytope with some facets removed and we relate it to the descents of permutations. Our result generalizes that of Early, Kim, and Li for hypersimplices.

Mathematics Subject Classifications: 05B35

Keywords: Positroid, Ehrhart theory