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The immanants of combinatorial matrices have many significant properties, including m-positivity and Schur positivity. While the immanants of Jacobi-Trudi matrices are known to be both m-positive and Schur positive, those of Giambelli…
Let $\Delta \subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_{\Delta}^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |k\Delta \cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients.…
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…
Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections…
A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^*$-vector of a spanning lattice polytope has no…
Reflexive polytopes have been studied from viewpoints of combinatorics, commutative algebra and algebraic geometry. A nef-partition of a reflexive polytope $\mathcal{P}$ is a decomposition $\mathcal{P}=\mathcal{P}_1+\cdots+\mathcal{P}_r$…
Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the…
We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive…
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side…
The matching polytope of a graph $G$ is the convex hull of the indicator vectors of the matchings on $G$. We characterize the graphs whose associated matching polytopes are Gorenstein, and then prove that all Gorenstein matching polytopes…
The Ehrhart polynomial of a lattice polytope $P$ encodes information about the number of integer lattice points in positive integral dilates of $P$. The $h^\ast$-polynomial of $P$ is the numerator polynomial of the generating function of…
Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator…
We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $\beta$-Grothendieck polynomials…
The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a…
We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and…
Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper…
We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials, as…
Graph polytopes arising from vertex-weighted graphs were first introduced by B\'ona, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is…
For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…
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…