  The paper establishes a connection between two recent combinatorial
developments in free probability: the non-crossing linked partitions introduced
by Dykema in 2007 to study the S-transform, and the partial order << on NC(n)
introduced by Belinschi and Nica in 2008 in order to study relations between
free and Boolean probability. More precisely, one has a canonical bijection
between NCL(n) (the set of all non-crossing linked partitions of {1, ..., n})
and the set {(p,q) | p,q in NC(n), p<<q}. As a consequence of this bijection,
one gets an alternative description of Dykema's formula expressing the moments
of a noncommutative random variable a in terms of the coefficients of the
reciprocal S-transform 1/S_a. Moreover, due to the Boolean features of <<, this
formula can be simplified to a form which resembles the moment-cumulant formula
from c-free probability.

Operator Algebras;Combinatorics