Marcelo Aguiar ; Aaron Lauve

Lagrange's Theorem for Hopf Monoids in Species
dmtcs:2887 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)

https://doi.org/10.46298/dmtcs.2887
Lagrange's Theorem for Hopf Monoids in Species
Authors: Marcelo Aguiar ; Aaron Lauve
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Marcelo Aguiar;Aaron Lauve
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 settheoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence.