Free cumulants, Schröder trees, and operads
arXiv:1604.04759 · doi:10.1016/j.aam.2017.02.002
Abstract
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder 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.
23 pages