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

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.

18 pages, 5 figures. Minor changes. Accepted for publication in the Electronic Journal of Combinatorics

References in corpus (6)

• Hypergraphic polytopes: combinatorial properties and antipode
• Hopf monoids and generalized permutahedra
• Combinatorial Hopf Algebras of Simplicial Complexes
• Cancelation free formula for the antipode of linearized Hopf monoid
• Hopf algebras and Tutte polynomials
• Polynomial invariants and reciprocity theorems for the Hopf monoid of hypergraphs and its sub-monoids

Cited by in corpus (1)

• Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra