Non-crossing linked partitions, the partial order $\ll$ on $NC(n)$, and the $S$-transform

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 $\ll$ 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 $\{ 1, \ldots , n \}$) and the set $\{ ( \alpha , \beta ) \mid \alpha , \beta \in NC(n), \alpha \ll \beta \}$. As a consequence of this bijection, one gets an alternative description of Dykema's formula expressing the moments of a non-commutative random variable $a$ in terms of the coefficients of the reciprocal $S$-transform $1/S_a$. Moreover, due to the Boolean features of $\ll$, this formula can be simplified to a form which resembles the moment-cumulant formula from $c$-free probability.