  We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or
cofree) and equipped with a given isomorphism to the free algebra over the
indecomposables (resp. the cofree coalgebra over the primitives). The choice of
such an isomorphism implies the existence a finer algebraic structure on the
Hopf algebra and on the indecomposables (resp. the primitives). For instance a
cofree-cocommutative right-sided combinatorial Hopf algebra is completely
determined by its primitive part which is a pre-Lie algebra. The key example is
the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf
algebra types gives rise to several good triples of operads. It involves the
operads: dendriform, pre-Lie, brace, and variations of them.
