Combinatorial expressions of Hopf polynomial invariants

Abstract

In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. The polynomials constructed this way are often subject to reciprocity theorems depending on the antipode of the associated Hopf monoid, i.e. while they are defined over positive integers, it is possible to find them a combinatorial interpretation over negative integers. In the same article Aguiar and Ardila then give a cancellation-free grouping-free formula for the antipode on generalized permutahedra and apply their constructions over some examples. In this work, we give a combinatorial interpretation of these polynomials over both positive integers and negative integers for the Hopf monoids of generalized permutahedra and hypergraphs and for every character on these two Hopf monoids. In the case of hypergraphs, we present two different proofs for the interpretation on negative integers, one using Aguiar and Ardila's antipode formula and one similar to the way Aval et al. defined a chromatic polynomial for hypergraphs in arXiv:1806.08546. We then deduce similar results on other combinatorial objects including graphs, simplicial complexes and building sets.

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

“…In this section we compare our results to the polynomial invariants from Hopf monoids developed in [AA17,Kar22]. This paper was motivated by giving a geometric interpretation of the combinatorial reciprocity theorems in [AA17].…”

“…The special case of this theorem for k = 0, i.e., generic directions, was obtained by Aguiar and Ardila [AA17], and earlier by Billera, Jia, and Reiner [BJR09] in a slightly different language. The k = 0 case was also recently extended in [Kar22]. As shown for some examples in [AA17, Section 18] the application of such a result to the various subclasses of generalized permutahedra yields already known combinatorial reciprocity theorems for their related combinatorial structures such as matroid polynomials [BJR09], Bergmann polynomials of matroids and Stanley's famous reciprocity theorem for graph colorings [Sta73].…”

“…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), andKaraboghossian (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).…”

“…Aguiar and Ardila develop a Hopf monoid structure on the species of generalized permutahedra, work with polynomial invariants defined by characters, and apply their antipode formula to get the combinatorial interpretation of the reciprocity result for generalized permutahedra for k = 0 (Theorem 16, below) [AA17,Sections 16,17]. This method is also used in [Kar22]. The approach in [BJR09] is similar to the one by Aguiar and Ardila.…”

“…Our main tool is a vertex description of hypergraphic polytopes in terms of acyclic orientations of hypergraphs (Proposition 18). More recent work by Karaboghossian [Kar20,Kar22] presents a more general version of the combinatorial reciprocity result for hypergraphs and an alternative proof with similar techniques as we present in Section 3.4.…”