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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


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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
PII: S 0002-9939(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.