Pruned Inside-Out Polytopes, Combinatorial Reciprocity Theorems and Generalized Permutahedra

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 show (quasi-)polynomiality and reciprocity results for 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.