(Non)commutative Hopf algebras of trees and (quasi)symmetric functions

Abstract

The Connes–Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes–Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson–Schwinger equations.