Examples of Com-PreLie Hopf algebras

We gives examples of Com-PreLie bialgebras, that is to say bialgebras with a preLie product satisfying certain compatibilities. Three families are defined on shuffle algebras: one associated to linear endomorphisms, one associated to linear form, one associated to preLie algebras. We also give all graded preLie product on K[X], making this bialgebra a Com-PreLie bialgebra, and classify all connected cocommutative Com-PreLie bialgebras.