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Séminaire Lotharingien de Combinatoire, B72d (2015), 22 pp.

# Bérénice Oger

# Incidence Hopf Algebra of the Hypertree Posets

**Abstract.**
We adapt the computation of characters on incidence Hopf algebras
introduced by Schmitt in the 1990s for families of bounded posets to a
family mixing bounded and unbounded finite posets. This computation
relies on the introduction of an auxiliary bialgebra: the coproduct in
this bialgebra enables us to compute the convolution of some
characters on the incidence Hopf algebra. After establishing a general
result on the link between the bialgebra and the incidence Hopf
algebra, we apply it to the family of hypertree posets and partition
posets.
This link for hypertree posets enables us to recover the Möbius
numbers of these posets due to the coproduct in the associated
bialgebra. This coproduct is computed using the number of hypertrees
with fixed valency set and fixed edge sizes set.

Received: March 18, 2014.
Accepted: May 25, 2015.

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