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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 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 g
From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum funct
Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $bf H$. That is, $H={mathcal K} ({bf H})$ or $H=overline{mathcal K} ({bf H})$. Unlike the functor $overline{mathcal K}$, the functor ${mathcal
A $d$-partite hypergraph is called fractionally balanced if there exists a non-negative function on its edge set that has constant degrees in each vertex side. Using a topological version of Halls theorem we prove lower bounds on the matching number
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobjec