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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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