Combinatorial properties of the noncommutative Faà di Bruno algebra
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 243-273.

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Summary: We give a new combinatorial interpretation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faà di Bruno algebra.
Keywords: noncommutative symmetric functions, combinatorial Hopf algebras, Lagrange inversion
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     author = {Bultel, Jean-Paul},
     title = {Combinatorial properties of the noncommutative {Fa\`a} di {Bruno} algebra},
     journal = {Journal of Algebraic Combinatorics},
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Bultel, Jean-Paul. Combinatorial properties of the noncommutative Faà di Bruno algebra. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 243-273. http://geodesic.mathdoc.fr/item/JAC_2013__38_2_a10/