Noncommutative free cumulants
Résumé
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.