Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra

TL;DR

This paper introduces pruned inside-out polytopes, a new geometric framework that generalizes existing reciprocity theorems for generalized permutahedra and hypergraph colorings, linking combinatorics and geometry.

Contribution

It defines pruned inside-out polytopes and extends reciprocity theorems to this new class, providing geometric proofs for hypergraph coloring reciprocity.

Findings

Generalized reciprocity theorems for permutahedra and hypergraphs

A geometric proof for hypergraph coloring reciprocity

Application of Ehrhart theory to pruned inside-out polytopes

Abstract

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck--Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017), Billera-Jia-Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). This proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.