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

Abstract

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 S -transform, and the partial order ≪ \ll on N C ( n ) 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 N C L ( n ) NCL(n) (the set of all non-crossing linked partitions of { 1 , … , n } \{ 1, \ldots , n \} ) and the set { ( α , β ) ∣ α , β ∈ N C ( n ) , α ≪ β } \{ ( \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 a in terms of the coefficients of the reciprocal S S -transform 1 / S a 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 c -free probability.