Geodesic
Inversion of Integral Series Enumerating Planar Trees
Séminaire lotharingien de combinatoire, Tome 53 (2005-2006)

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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. 

@article{SLC_2005-2006_53_a3,
     author = {Jean-Louis Loday},
     title = {Inversion of {Integral} {Series} {Enumerating} {Planar} {Trees}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {53},
     year = {2005-2006},
     url = {http://geodesic.mathdoc.fr/item/SLC_2005-2006_53_a3/}
}
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UR  - http://geodesic.mathdoc.fr/item/SLC_2005-2006_53_a3/
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Jean-Louis Loday. Inversion of Integral Series Enumerating Planar Trees. Séminaire lotharingien de combinatoire, Tome 53 (2005-2006). http://geodesic.mathdoc.fr/item/SLC_2005-2006_53_a3/