A matroidal twist on a formula of Brion
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.
Cite
@article{arxiv.2604.15253,
title = {A matroidal twist on a formula of Brion},
author = {Matthias Beck and Caroline Klivans and Dustin Ross},
journal= {arXiv preprint arXiv:2604.15253},
year = {2026}
}
Comments
minor revisions, references added, 24 pages, 7 figures, comments welcome