Polynomial invariants and reciprocity theorems for the Hopf monoid of hypergraphs and its sub-monoids

TL;DR

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

Contribution

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

Findings

New polynomial invariant for hypergraphs

Reciprocity theorem for hypergraph invariants

Unified approach to invariants of graphs, complexes, and building sets

Abstract

In arXiv:1709.07504 Aguiar and Ardila 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.