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Séminaire Lotharingien de Combinatoire, 78B.13 (2017), 12 pp.

# Carolina Benedetti and Nantel Bergeron

# The Antipode of Linearized Hopf Monoids

**Abstract.**
Many combinatorial Hopf algebras *H* in the literature are the
functorial image of a linearized Hopf monoid **H**. That is
*H* = **K**(**H**) or
*H* = **K**^{-}(**H**).
For the functor **K** the antipode of **H** may not
be preserved, but the Hopf monoid
**L** x **H** gives
*H* = **K**(**H**) =
**K**^{-}(**L** x
**H**) and the functor
**K**^{-} preserves antipodes. In
this paper, we give a cancelation free and multiplicity free formula
for the antipode of
**L** x **H**. We also compute the
antipode for **H** when it is commutative and cocommutative. We
get new formulas that are not always cancelation free but can be used
to obtain one for *H* in some cases. The formulas for **H**
involve acyclic orientations of hypergraphs. In an example, we
introduce a chromatic invariant for the increasing sequences of a
permutation and show that its evaluation at *t* = -1 relates to another
statistic on permutations.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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