Related papers: Ehrhart polynomials of partial permutohedra

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in…

Let $P_{\lambda\Sigma_n}$ be the Ehrhart polynomial associated to an intergal multiple $\lambda$ of the standard symplex $\Sigma_n \subset \mathbb{R}^n$. In this paper we prove that if $(M, L)$ is an $n$-dimensional polarized toric manifold…

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap…

Ehrhart polynomials are extensively-studied structures that interpolate the discrete volume of the dilations of integral $n$-polytopes. The coefficients of Ehrhart polynomials, however, are still not fully understood, and it is not known…

This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real…

The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart…

Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to…

In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn…

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…

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize…

Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $\lambda : A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $\lambda$.…

We characterize all signed Minkowski sums that define generalized permutahedra, extending results of Ardila-Benedetti-Doker (2010). We use this characterization to give a complete classification of all positive, translation-invariant,…

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…

A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a…

Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, .... We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups, expressing them…

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart…

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…

In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart…

Given a rational polytope $P \subset \mathbb R^d$, the numerical function counting lattice points in the integral dilations of $P$ is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial $\mathrm{ehr}_P$ of $P$. In this…

We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in $\mathbb{R}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset…