Combinatorial properties of the noncommutative Faà di Bruno algebra. (Q364703)

The Brouder-Frabetti-Krattenthaler and Gessel-Pak-Postnikov-Retakh formulas for the antipode of the noncommutative Faà di Bruno are noncommutative versions of the classical Lagrange inversion formula for formal series in one variable. The present text gives a combinatorial interpretation of these formulas. Namely, the antipode of a simple complete noncommutative symmetric function is expanded in the basis of ribbons: the appearing coefficients are described in terms of nondecreasing parking functions satisfying certain conditions. The deformation of the noncommutative Faà di Bruno Hopf algebra \(H_{FdB}\) is generically isomorphic to \(H_{FdB}\). An explicit isomorphism is here given, and combinatorial interpretations of the antipode of these deformations are deduced from the existence of an algebra morphism between the deformation of the classical object.