相关论文: On the Ehrhart Polynomial of Schubert Matroids

In this paper, we prove that the Chow ring and augmented Chow ring of a matroid are equivariantly $\gamma$-positive under the action of any group of automorphisms. Our approach provides an explicit combinatorial interpretation of 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 $N$…

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that $s(n) \sim m(n)$, where $m(n)$ denotes the number of matroids on a fixed groundset of size $n$, and $s(n)$ the number of sparse paving matroids. In…

Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino…

We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a…

A matroid is uniform if and only if it has no minor isomorphic to $U_{1,1}\oplus U_{0,1}$ and is paving if and only if it has no minor isomorphic to $U_{2,2}\oplus U_{0,1}$. This paper considers, more generally, when a matroid $M$ has no…

In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and…

Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the…

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…

We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main…

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In math.CO/0402148, the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial…

A half-integral polygon with quasi-period collapse behaves similarly to a lattice polygon in the sense that the number of lattice points in its integer dilates can be calculated as values of a polynomial, its Ehrhart polynomial. As a main…

For an $n$-dimensional lattice simplex $\Delta_{(1,\mathbf{q})}$ with vertices given by the standard basis vectors and $-\mathbf{q}$ where $\mathbf{q}$ has positive entries, we investigate when the Ehrhart $h^*$-polynomial for…

In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very…

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of…

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of…

We use counting arguments to show that asymptotically almost all sparse paving matroids contain an $H$-minor, where $H$ falls into one of several simple classes of matroids. Furthermore the result holds for all $H$ in a larger class of…

We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe the Ehrhart quasi-polynomial of a rational…

Motivated by the question of whether Chow polynomials of matroids have only real roots, this article revisits the known relationship between Eulerian polynomials and the Hilbert series of Chow rings of permutohedral varieties. This is done…