A combinatorial interpretation for two transformations of series that commute with compositional inversion

Abstract

For a formal power series g(t) = 1 [1 + ∑ n=1 ∞h nt n] with nonnegative integer coefficients, the compositional inverse f(t) = t · f(t) of g(t) = t · g(t) is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of h i colors. Since the compositional inverse of the Euler transformation of f( t) is the star transformation [[ g( t)] −1 − 1] −1 of g( t), [2], it follows that the Euler transformation of f( t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of h i colors for i > 1, and h 1 − 1 colors for i = 1.