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Bialgebras in cointeraction, the antipode and the eulerian idempotent
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We give here a review of results about double bialgebras, that is to say bialgebras with two coproducts, the first one being a comodule morphism for the coaction induced by the second one. An accent is put on the case of connected bialgebras. The subjects of these results are the monoid of characters and their actions, polynomial invariants, the antipode and the eulerian idempotent. As examples, they are applied on a double bialgebra of graphs and on quasishuffle bialgebras. This includes a new proof of a combinatorial interpretation of the coefficients of the chromatic polynomial due to Greene and Zaslavsky.
Forward citations
Cited by 2 Pith papers
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An equality for balanced digraphs
The number of k-arc acyclic subdigraphs in which every vertex can reach a fixed root is independent of the root, for any balanced digraph.
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Extended generalized permutahedra, and cointeracting bialgebras
EGPs and affine-cone EGPs form cointeracting bimonoids in species measured by the monoid of EGPs, via the face-and-tangent-cone map, with explicit submodular functions for faces and cones.
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