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

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.

12 page, 5 figures. To appear in the Proceedings of Fpsac 2019. This article draws heavily from arXiv:1806.08546

References in corpus (2)

• Hypergraphic polytopes: combinatorial properties and antipode
• Cancelation free formula for the antipode of linearized Hopf monoid

Cited by in corpus (1)

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