An antipode formula for the natural Hopf algebra of a set operad
A set-operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a set-operad, we give here a simple construction of a Hopf algebra that we call {\em the natural Hopf algebra} of the operad. We obtain a combinatorial formula for its antipode in terms of Shr\"oder trees, generalizing the Hayman-Schmitt formula for the Fa\'a di Bruno Hopf algebra. From there we derive more readable 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 an incidence reduced 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.