The Hopf Monoid of Hypergraphs and its Sub-Monoids: Basic Invariant and Reciprocity Theorem

  • Jean-Christophe Aval
  • Théo Karaboghossian
  • Adrian Tanasa


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.

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