A short proof of the Lagrange-Good formula

Since Good’s paper [5j appeared in 1960, :i lot of other works have been published on this theme. Good’s proof of his theorem being analytical, <:he later authors considered Lagrange inversion within the theory of formal power series. The first attempt for a purely combinatorial proof was made by Chottin [ 1 j, who treated a special case of the two dimensional formula. Tutte gave an extensive development of this subject in [ll j for an arbitrary number of variabks. While studying Rota’s theory of polynomial sequences of binomial type (see [8,9, lo]), Cigler [2], Garsia and Joni [3 j realized its intimate connection with Lagrange inversion (in the one dimensional case); &neralizing Rota’v theory to higher dimensions they obtained an analogot:s setting for the Lagrange-Good formula (see [2,4,6,73). This paper is mainly based on these work, it follows the same ideas and gives only a simplified version of the more technical part of their proofs. Readers interested in the connection to Rota’s theory which will not be treated here are referred to the above cited papers.