Lagrange Inversion for Species

1. Introduction. The Lagrange inversion formula is one of the fundamental results of enumerative combinatorics. It expresses the coefficients of powers of the compositional inverse of a power series in terms of the coefficients of powers of the original power series. G. Labelle [10] extended Lagrange inversion to cycle index series, which are equivalent to symmetric functions. Although motivated by Joyal's theory of species of structures [7], Labelle's proof was algebraic, and was based on the ordinary multivariable Lagrange inversion formula. We give here a new proof of this formula in the context of the theory of species. In contrast with the proof given in [10], the bijections involved are all natural in the categorical sense of the word. Our approach involves several new or little-known operations on species, some of which were studied earlier by Joyal [9], and which have other enumerative applications.