Polynomial realizations of Hopf algebras built from nonsymmetric operads

The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using alphabets of noncommutative variables endowed with unary and binary relations. By using particular alphabets, we establish links between $\mathbf{N} \cdot \mathcal{O}$ 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.