The Hopf monoid of hypergraphs and its sub-monoids: basic invariant and reciprocity theorem

TL;DR

This paper introduces a new polynomial invariant for hypergraphs within a Hopf monoid framework, providing a combinatorial reciprocity theorem and unifying various known invariants across combinatorial structures.

Contribution

It defines a novel polynomial invariant on hypergraphs, interprets it combinatorially at negative integers, and recovers existing invariants and reciprocity theorems for related combinatorial objects.

Findings

Introduces a new hypergraph invariant within a Hopf monoid structure

Provides a combinatorial reciprocity theorem for hypergraphs

Unifies known invariants for graphs, complexes, and building sets

Abstract

In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.