Séminaire Lotharingien de Combinatoire, B53d (2005), 16 pp.

Jean-Louis Loday

Inversion of Integral Series Enumerating Planar Trees

Abstract. We consider an integral series f(X,t) which depends on the choice of a set X of labelled planar rooted trees. We prove that its inverse with respect to composition is of the form f(Z,t) for another set Z of trees, deduced from X. The proof is self-contained, though inspired by the Koszul duality theory of quadratic operads. In the same vein we give a conceptual proof for the formulas giving the coefficients of the inverse with respect to composition of the generic formal power series.

Received: November 15, 2004. Accepted: November 11, 2005. Final Version: November 23, 2005.

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