Non-crossing linked partitions, the partial order on , and the -transform

Author:
Alexandru Nica

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1273-1285

MSC (2010):
Primary 46L54; Secondary 05A18

Published electronically:
December 1, 2009

MathSciNet review:
2578521

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Abstract | References | Similar Articles | Additional Information

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 -transform, and the partial order on 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 (the set of all non-crossing linked partitions of ) and the set . As a consequence of this bijection, one gets an alternative description of Dykema's formula expressing the moments of a non-commutative random variable in terms of the coefficients of the reciprocal -transform . Moreover, due to the Boolean features of , this formula can be simplified to a form which resembles the moment-cumulant formula from -free probability.

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Additional Information

**Alexandru Nica**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Email:
anica@math.uwaterloo.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10218-6

Received by editor(s):
January 29, 2009

Published electronically:
December 1, 2009

Additional Notes:
Research supported by a Discovery Grant from NSERC, Canada

Communicated by:
Marius Junge

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.