A matroidal twist on a formula of Brion

TL;DR

This paper extends Brion's formula for lattice points in polytopes to a matroidal setting, revealing recursive, reciprocal properties and connecting to matroid Euler characteristics.

Contribution

It introduces a modified rational function sum for generalized permutohedra based on matroids, providing new combinatorial insights and connections.

Findings

The Laurent polynomial Q_M(P) mimics lattice point behavior in P.

Q_M(P) exhibits recursive and reciprocity properties.

Evaluating Q_M(P) at 1 yields the matroid Euler characteristic.

Abstract

Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron. We study a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we describe how the resulting Laurent polynomial Q_M(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating Q_M(P) at 1, we recover the matroid Euler characteristic of Larson, Li, Payne, and Proudfoot, so the combinatorial approach in this paper gives new insight into studying these quantities.