Ehrhart Polynomials of Matroid Polytopes and Polymatroids

Abstract

Due to a production error the article was published with mistakes in the paragraph following Theorem 1. The corrected paragraph appears below.The computation of volumes is one of the most fundamental geometric operations and it has been investigated by several authors from the algorithmic point of view. Although there are a few cases for which the volume can be computed efficiently (e.g., for convex polytopes in fixed dimension), it has been proved that computing the volume of polytopes of varying dimension is #P -hard [9, 17, 26, 30]. Moreover it was proved that even approximating the volume is hard [19]. Clearly, computing Ehrhart polynomials is a harder problem still. To our knowledge there were only two previously known families of varying-dimension polytopes for which there is efficient computation of the volume. These two families are simplices or simple polytopes with a polynomial number of vertices (this follows from Lawrence's volumeThe online version of the original article can be found under

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

“…We then use the latter and some known results on the Ehrhart polynomial of order polytopes in order to prove unimodality of the h * -vector for two infinite families of snakes (Theorem 4.9). This provides new evidence for a challenging conjecture of De Loera, Haws, and Köppe [11] which was only known to hold for the class of rank two uniform matroids and for a finite list of examples. In Section 5, we end by discussing further cases in which this conjecture holds.…”

On Positivity of Ehrhart Polynomials

“…In [31], De Loera, Haws, and Koeppe study the Ehrhart polynomials of matroid base polytopes, and conjecture those all have positive coefficients. However, it turns out that every matroid base polytope is a generalized permutohedron [1,Section 2].…”

“…However, it follows from work by Ardila, Benedetti and Doker that type-Y generalized permutohedra do not contain all matroid base polytopes [1, Proposition 2.3 and Example 2.6]. Therefore, Corollary 3.1.5 does not settle either Conjecture 3.1.1 or the Ehrhart positivity conjecture on matroid base polytopes by De Loera et al[31].…”

Facets of the r-Stable (n, k)-Hypersimplex

“…Generalizing the standard (n − 1)-simplex, the (n, k)-hypersimplices serve as a useful collection of examples in these various contexts. While these polytopes are well studied, there remain interesting open questions about their properties in the field of Ehrhart theory, the study of integer point enumeration in dilations of rational polytopes (see, for example, [4]). The r-stable (n, k)-hypersimplices are a collection of lattice polytopes within the (n, k)-hypersimplex that were introduced in [2] for the purpose of studying unimodality of the Ehrhart δ -polynomials of the (n, k)-hypersimplices.…”