Polynomial invariant and reciprocity theorem for the Hopf monoid of hypergraphs

Hop monoids were introduced by Aguiar and Mahajan and use algebra to formalise the notions of merging and spliting of combinatorial objects (words concatenation, graph restriction etc). Aguiar and Ardila have shown a useful application of this formalism in defining and computing polynomial invariants. In this talk I present the notion of Hopf monoid and use the works of Aguiar and Ardila to define a new polynomial invariant on hypergraphs. I then show that this invariant is subject to a reciprocity theorem similar to Stanley’s reciprocity theorem on the chromatic polynomial of a graph. Finally I show how similar results on other combinatorial objects can be found as a consequence of this theorem.