Free cumulants, Schröder trees, and operads

Abstract: The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.