An antipode formula for the natural Hopf algebra of a set operad

A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its antipode in terms of Schröder trees, generalizing the Haiman–Schmitt formula for the Faá di Bruno Hopf algebra. From there we derive antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli–Thibon combinatorial form of the antipode for the non-commutative Hopf algebra of formal diffeomorphisms.