The infinitesimal Hopf algebra and the poset of planar forests

TL;DR

This paper introduces an infinitesimal Hopf algebra for planar trees, extending the Connes-Kreimer algebra, and explores its combinatorial structure through posets and M"obius inversion.

Contribution

It generalizes the non-commutative Connes-Kreimer Hopf algebra by defining a new infinitesimal Hopf algebra with a combinatorial interpretation.

Findings

Defined a non-degenerate pairing and dual basis.

Connected the algebraic structure to the Tamari poset.

Provided a method to compute the dual basis via M"obius inversion.

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 M\"obius inversion.