Sophie Rehberg
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) and Billera-Jia-Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval--Karaboghossian--Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.