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تسجيل مستخدم جديدThe Hopf monoid of hypergraphs and its sub-monoids: basic invariant and reciprocity theorem
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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.
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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 g
ive 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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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 ${mathcal
K}$ applied to ${bf H}$ may not preserve the antipode of ${bf H}$. In this case, one needs to consider the larger Hopf monoid ${bf L}times{bf H}$ to get $H={mathcal K} ({bf H})=overline{mathcal K}({bf L}times{bf H})$ and study the antipode in ${bf L}times{bf H}$. One of the main results in this paper provides a cancelation free and multiplicity free formula for the antipode of ${bf L}times{bf H}$. From this formula we obtain a new antipode formula for $H$. We also explore the case when ${bf H}$ is commutative and cocommutative. In this situation we get new antipode formulas that despite of not being cancelation free, can be used to obtain one for $overline{mathcal K}({bf H})$ in some cases. We recover as well many of the well-known cancelation free formulas in the literature. One of our formulas for computing the antipode in ${bf H}$ involves acyclic orientations of hypergraphs as the central tool. In this vein, we obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanleys (-1)-color theorem. One of our examples introduces a {it chromatic} polynomial for permutations which counts increasing sequences of the permutation satisfying a pattern. We also study the statistic obtained after evaluating such polynomial at $-1$. Finally, we sketch $q$ deformations and geometric interpretations of our results. This last part will appear in a sequel paper in joint work with J. Machacek.
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