Noncrossing linked partitions, the partial order on , and the transform
Author:
Alexandru Nica
Journal:
Proc. Amer. Math. Soc. 138 (2010), 12731285
MSC (2010):
Primary 46L54; Secondary 05A18
Published electronically:
December 1, 2009
MathSciNet review:
2578521
Fulltext PDF Free Access
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Abstract: The paper establishes a connection between two recent combinatorial developments in free probability: the noncrossing 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 noncrossing 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 noncommutative 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 momentcumulant formula from free probability.
 1.
Serban
T. Belinschi and Alexandru
Nica, 𝜂series and a Boolean BercoviciPata bijection for
bounded 𝑘tuples, Adv. Math. 217 (2008),
no. 1, 1–41. MR 2357321
(2009c:46088), 10.1016/j.aim.2007.06.015
 2.
Serban
T. Belinschi and Alexandru
Nica, Free Brownian motion and evolution towards ⊞infinite
divisibility for 𝑘tuples, Internat. J. Math.
20 (2009), no. 3, 309–338. MR 2500073
(2010g:46108), 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
(98j:46069)
 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
(2009e:05026), 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
(2008k:46193), 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
(2008k:46198)
 8.
M. Popa. Noncrossing linked partitions and multiplication of free random variables, preprint, 2008, arXiv:0812.2064.
 9.
Rodica
Simion, Noncrossing partitions, Discrete Math.
217 (2000), no. 13, 367–409 (English, with
English and French summaries). Formal power series and algebraic
combinatorics (Vienna, 1997). MR 1766277
(2001g:05011), 10.1016/S0012365X(99)002733
 10.
Roland
Speicher, Multiplicative functions on the lattice of noncrossing
partitions and free convolution, Math. Ann. 298
(1994), no. 4, 611–628. MR 1268597
(95h:05012), 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 GianCarlo Rota and appendix 1 by
Sergey Fomin. MR
1676282 (2000k:05026)
 12.
Dan
Voiculescu, Multiplication of certain noncommuting random
variables, J. Operator Theory 18 (1987), no. 2,
223–235. MR
915507 (89b:46076)
 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
(94c:46133)
 1.
 S.T. Belinschi, A. Nica. series and a Boolean BercoviciPata bijection for bounded tuples, Advances in Mathematics 217 (2008), 141. MR 2357321 (2009c:46088)
 2.
 S.T. Belinschi, A. Nica. Free Brownian motion and evolution towards infinite divisibility for tuples, International Journal of Mathematics 20 (2009), 309338. MR 2500073
 3.
 M. Bozejko, M. Leinert, R. Speicher. Convolution and limit theorems for conditionally free random variables, Pacific Journal of Mathematics 175 (1996), 357388. MR 1432836 (98j:46069)
 4.
 W.Y.C. Chen, S.Y.J. Wu, C.H. Yan. Linked partitions and linked cycles, European Journal of Combinatorics 29 (2008), 14081426. MR 2423730 (2009e:05026)
 5.
 K. Dykema. Multilinear function series and transforms in free probability theory, Advances in Mathematics 208 (2007), 351407. MR 2304321 (2008k:46193)
 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.
 A. Nica, R. Speicher. Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, 2006. MR 2266879 (2008k:46198)
 8.
 M. Popa. Noncrossing linked partitions and multiplication of free random variables, preprint, 2008, arXiv:0812.2064.
 9.
 R. Simion. Noncrossing partitions, Discrete Mathematics 217 (2000), 367409. MR 1766277 (2001g:05011)
 10.
 R. Speicher. Multiplicative functions on the lattice of noncrossing partitions and free convolution, Mathematische Annalen 298 (1994), 611628. MR 1268597 (95h:05012)
 11.
 R.P. Stanley. Enumerative combinatorics, volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999. MR 1676282 (2000k:05026)
 12.
 D. Voiculescu. Multiplication of certain noncommuting random variables, Journal of Operator Theory 18 (1987), 223235. MR 915507 (89b:46076)
 13.
 D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables, CRM Monograph Series, 1, American Mathematical Society, 1992. MR 1217253 (94c:46133)
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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/S0002993909102186
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.
