It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. For many examples, see David Vella’s paper. The resulting identities involve partitions of integers. I imagine that it should be possible to get a connection with free probability theory in which a noncommutative Faa di Bruno formula gives identities involving noncrossing partitions.
Question: Is there a free-probability analogue of the Faa-di-Bruno formula that naturally generates combinatorial identities involving non-crossing partitions?
I write naturally here because I'm interested in seeing something as close as possible to what has been done in Vella's paper. (This may be a tall order.)
In this paper of Brouder, Fabretti and Krattenthaler, a noncommutative co-commutative Hopf algebra is constructed whose abelianization gives precisely the Faa di Bruno Hopf algebra, from which one can recover the classical Faa di Bruno formula. Basically, one considers one-variable generating functions of an infinite sequence of algebraically free coefficients. Also, in this case an explicit formula for the antipode is given in which Catalan numbers appear, and is interpreted in terms of trees and algebra free products. I'm not certain if this may provide a good starting point, should what I'm looking for not already exist in the literature.
Along these lines, there is the paper of Anschelevich, Effros and Popa here.
EDIT: Michael Hardy's paper linked to below emphasizes instead partitions of sets, which allows for the computation of cumulants. This indicates further that there may be a connection, since noncrossing partitions appear precisely when considering free cumulants. Here's a paper of Krawczyk and Speicher discussing the combinatorics of free cumulants.