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Journal Articles Journal of Algebraic Combinatorics Year : 2009

The infinitesimal Hopf algebra and the poset of planar forests

Loïc Foissy

Abstract

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.
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hal-00238498 , version 1 (04-02-2008)

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Loïc Foissy. The infinitesimal Hopf algebra and the poset of planar forests. Journal of Algebraic Combinatorics, 2009. ⟨hal-00238498⟩

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