Polynomial realizations of Hopf algebras built from nonsymmetric operads

The natural Hopf algebra N · O of an operad O is a Hopf algebra whose bases are indexed by some words on O. We construct polynomial realizations of N · O by using alphabets of noncommutative variables endowed with unary and binary relations. By using particular alphabets, we establish links between N · O and some other Hopf algebras including the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the noncommutative Connes-Kreimer Hopf algebra of Foissy, the noncommutative Fa\`a di Bruno Hopf algebra and its deformations, the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon, and the double tensor Hopf algebra of Ebrahimi-Fard and Patras.