Fa\`a di Bruno for operads and internal algebras

For any coloured operad R, we prove a Fa`a di Bruno formula for the connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Fa\a di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faa di Bruno formula for P-trees of G'alvez–Kock–Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following G'alvez–Kock–Tonks, we work at the objective level of groupoid slices, hence all proofs are bijective': the formula is established as the homotopy cardinality of an explicit equivalence of groupoids. In fact we establish the formula more generally in a relative situation, for algebras for one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Fa\a di Bruno formula of Brouder–Frabetti–Krattenthaler).