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

TL;DR

This paper links non-crossing linked partitions and the partial order << in free probability, providing a new bijection and simplifying formulas related to the S-transform and moments of noncommutative random variables.

Contribution

It introduces a canonical bijection between non-crossing linked partitions and pairs in NC(n) ordered by <<, offering an alternative formula for moments in free probability.

Findings

Established a bijection between NCL(n) and pairs (p,q) with p<<q in NC(n)

Provided an alternative description of Dykema's moment formula involving the reciprocal S-transform

Simplified the moment formula to resemble c-free probability formulas

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-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.