Related papers: Ehrhart polynomials of matroid polytopes and polym…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
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…
Over a decade ago De Loera, Haws and K\"oppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding $h^*$-polynomials form a unimodal sequence. The first of…
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all…
This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as…
The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We…
In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove…
De Loera et al. 2009, showed that when the rank is fixed the Ehrhart polynomial of a matroid polytope can be computed in polynomial time when the number of elements varies. A key to proving this is the fact that the number of simplicial…
We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and…
We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the $\beta$-invariant of the matroid. This yields a lattice-point counting formula for the…
There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$…
We give a combinatorial formula for the Ehrhart coefficients of a certain class of weighted multi-hypersimplices. In a special case, where these polytopes coincide with the base polytope of the panhandle matroid $\textrm{Pan}_{k,n-2,n}$, we…
We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of…
We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear…
Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…
Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if…
We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be…
There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to…