Polynomial realizations of natural Hopf algebras of nonsymmetric operads

The natural Hopf algebra $\mathcal{N}\mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We introduce new bases of these Hopf algebras deriving from free operads via new lattice structures on their basis elements. We construct polynomial realizations of $\mathcal{N}\mathcal{O}$ by using alphabets of variables endowed with unary and binary relations. By specializing our polynomial realizations, we discover links between $\mathcal{N}\mathcal{O}$ and the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the Connes-Kreimer Hopf algebra of Foissy, the Faà di Bruno Hopf algebra and its deformations, and the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon.