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

Full-text PDF Free Access

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.

**1.**Serban T. Belinschi and Alexandru Nica,*𝜂-series and a Boolean Bercovici-Pata bijection for bounded 𝑘-tuples*, Adv. Math.**217**(2008), no. 1, 1–41. MR**2357321**, 10.1016/j.aim.2007.06.015**2.**Serban T. Belinschi and Alexandru Nica,*Free Brownian motion and evolution towards ⊞-infinite divisibility for 𝑘-tuples*, Internat. J. Math.**20**(2009), no. 3, 309–338. MR**2500073**, 10.1142/S0129167X09005303**3.**Marek Bożejko, Michael Leinert, and Roland Speicher,*Convolution and limit theorems for conditionally free random variables*, Pacific J. Math.**175**(1996), no. 2, 357–388. MR**1432836****4.**William Y. C. Chen, Susan Y. J. Wu, and Catherine H. Yan,*Linked partitions and linked cycles*, European J. Combin.**29**(2008), no. 6, 1408–1426. MR**2423730**, 10.1016/j.ejc.2007.06.022**5.**Kenneth J. Dykema,*Multilinear function series and transforms in free probability theory*, Adv. Math.**208**(2007), no. 1, 351–407. MR**2304321**, 10.1016/j.aim.2006.02.011**6.**M. Mastnak, A. Nica. Hopf algebras and the logarithm of the -transform, to appear in*Transactions of the American Mathematical Society*, arXiv:0807.4169.**7.**Alexandru Nica and Roland Speicher,*Lectures on the combinatorics of free probability*, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR**2266879****8.**M. Popa. Non-crossing linked partitions and multiplication of free random variables, preprint, 2008, arXiv:0812.2064.**9.**Rodica Simion,*Noncrossing partitions*, Discrete Math.**217**(2000), no. 1-3, 367–409 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR**1766277**, 10.1016/S0012-365X(99)00273-3**10.**Roland Speicher,*Multiplicative functions on the lattice of noncrossing partitions and free convolution*, Math. Ann.**298**(1994), no. 4, 611–628. MR**1268597**, 10.1007/BF01459754**11.**Richard P. Stanley,*Enumerative combinatorics. Vol. 2*, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR**1676282****12.**Dan Voiculescu,*Multiplication of certain noncommuting random variables*, J. Operator Theory**18**(1987), no. 2, 223–235. MR**915507****13.**D. V. Voiculescu, K. J. Dykema, and A. Nica,*Free random variables*, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR**1217253**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
46L54,
05A18

Retrieve articles in all journals with MSC (2010): 46L54, 05A18

Additional Information

**Alexandru Nica**

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

Email:
anica@math.uwaterloo.ca

DOI:
https://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.