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.
In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. The polynomials constructed this way are often subject to reciprocity theorems depending on the antipode of the associated Hopf monoid, i.e. while they are defined over positive integers, it is possible to find them a combinatorial interpretation over negative integers. In the same article Aguiar and Ardila then give a cancellation-free grouping-free formula for the antipode on generalized permutahedra and apply their constructions over some examples. In this work, we give a combinatorial interpretation of these polynomials over both positive integers and negative integers for the Hopf monoids of generalized permutahedra and hypergraphs and for every character on these two Hopf monoids. In the case of hypergraphs, we present two different proofs for the interpretation on negative integers, one using Aguiar and Ardila's antipode formula and one similar to the way Aval et al. defined a chromatic polynomial for hypergraphs in arXiv:1806.08546. We then deduce similar results on other combinatorial objects including graphs, simplicial complexes and building sets.
We propose a new way of defining and studying operads on multigraphs and similar objects. For this purpose, we use the combinatorial species setting. We study in particular two operads obtained with our method. The former is a direct generalization of the Kontsevich-Willwacher operad. This operad can be seen as a canonical operad on multigraphs, and has many interesting suboperads. The latter operad is a natural extension of the pre-Lie operad in a sense developed here and it is related to the multigraph operad. We also present various results on some of the finitely generated suboperads of the multigraph operad and establish links between them and the commutative operad and the commutative magmatic operad.
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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