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.

Combinatorial Generation via Permutation Languages. V. Acyclic Orientations

“…Again, this definition does not change if we require the orientations h to be acyclic. A proof of these facts is implicit in previous works [BBM19] and spelled out by Rehberg [Reh21] (Proposition 3.9).…”

“…It can be shown that δ D is a vertex of Z(G) if and only if D is acyclic, hence the above definition remains valid if we restrict D to be an acyclic orientation of G. For a simple proof of this fact, we refer to that of a more general statement given as Proposition 3.9 in [Reh21]. The following lemma is a simple consequence of the definitions (see also the discussion in [SZ98]).…”

“…The general definition is most useful for a complete characterization of the faces of the hypergraphic polytope. Rehberg [Reh21] refers to our definition as a heading instead of an orientation. π yields the orientation h. The set {ext(P H,h ) | h ∈ AO H } is therefore a partition of S n into equivalence classes.…”