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Lagrange's Theorem for Hopf Monoids in Species

Abstract : We prove Lagrange's theorem for Hopf monoids in the category of connected species. We deduce necessary conditions for a given subspecies $\textrm{k}$ of a Hopf monoid $\textrm{h}$ to be a Hopf submonoid: each of the generating series of $\textrm{k}$ must divide the corresponding generating series of $\textrm{k}$ in ℕ〚x〛. Among other corollaries we obtain necessary inequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In the set-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence.
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Marcelo Aguiar, Aaron Lauve. Lagrange's Theorem for Hopf Monoids in Species. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.15-26, ⟨10.46298/dmtcs.2887⟩. ⟨hal-01215090⟩

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