Séminaire Lotharingien de Combinatoire, 78B.58 (2017), 12 pp.

Matthieu Josuat-Vergès, Frédéric Menous, Jean-Christophe Novelli and Jean-Yves Thibon

Noncommutative Free Cumulants

Abstract. The relation between moments and free cumulants in free probability is essentially a compositional inversion. We lift it at the level of the noncommutative Faà di Bruno algebra, and of an operad of Schröder trees. We get a new formula for free cumulants in terms of trees, and we recover an interpretation of the relation in terms of characters due to Ebrahimi-Fard and Patras.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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