Hopf algebras on planar trees and permutations

Abstract

We endow the space of rooted planar trees with an structure of Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labelled trees, $n$--trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto--Reutenauer and Loday--Ronco via planar rooted trees.