Multiplicative functions on the lattice of non-crossing partitions and free convolution. (English) Zbl 0791.06010
We examine the lattice of non-crossing partitions both from a combinatorial and from a probabilistic point of view. In the first part, we consider multiplicative functions on this lattice and describe the convolution of such functions by generating power series. This is used for deriving some known results on non-crossing partitions in a unified and short way. In the second part, we work out the connection between the lattice of non-crossing partitions and the ‘free convolution’ of probability measures in the sense of Voiculescu. A new combinatorial proof of Voiculescu’s main formula for this convolution is given.
Reviewer: R.Speicher (Heidelberg)
MSC:
| 06A07 | Combinatorics of partially ordered sets |
| 46L51 | Noncommutative measure and integration |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
| 60C05 | Combinatorial probability |
Keywords:
free convolution of probability measures; lattice of non-crossing partitions; multiplicative functions; generating power seriesReferences:
| [1] | Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. PAM-572 (Preprint 1992) · Zbl 0806.46070 |
| [2] | Bonin, J., Shapiro, L., Simion, R.: Someq-analogues of the Schr?der numbers arising from combinatorial statistics on lattice paths. (Preprint 1991) · Zbl 0783.05008 |
| [3] | Bourbaki, N.: Elements of mathematics; Algebra I. Berlin Heidelberg New York: Springer 1988 |
| [4] | Comtet, L.: Advanced combinatorics. Dordrecht: Reidel 1974 · Zbl 0283.05001 |
| [5] | Edelman, P.H.: Chain enumeration and non-crossing partitions. Discrete Math.31, 171-180 (1980) · Zbl 0443.05011 · doi:10.1016/0012-365X(80)90033-3 |
| [6] | Edelman, P.H.: Multichains, non-crossing partitions and trees. Discrete Math.40, 171-179 (1982) · Zbl 0496.05007 · doi:10.1016/0012-365X(82)90118-2 |
| [7] | Edelman, P.H., Simion, R.: Chains in the lattice of non-crossing partitions. (Preprint 1991) · Zbl 0795.05013 |
| [8] | Glockner, P., Sch?rmann, M., Speicher, R.: Realization of free white noises. Arch. Math.58, 407-416 (1992) · doi:10.1007/BF01189934 |
| [9] | Hilton, P., Pederson, J.: Catalan numbers, their generalization, and their uses. Math. Intell.13 (no. 2), 64-75 (1991) · Zbl 0767.05010 · doi:10.1007/BF03024089 |
| [10] | Klarner, D.A.: Correspondences between plane trees and binary sequences. J. Comb. Theory9, 401-411 (1970) · Zbl 0205.54702 · doi:10.1016/S0021-9800(70)80093-X |
| [11] | Kreweras, G.: Sur les partitions non crois?es d’un cycle. Discrete Math.1, 333-350 (1972) · Zbl 0231.05014 · doi:10.1016/0012-365X(72)90041-6 |
| [12] | Maassen, H.: Addition of freely independent random variables. J.Funct. Anal.106, 409-438 (1992) · Zbl 0784.46047 · doi:10.1016/0022-1236(92)90055-N |
| [13] | Polya, G., Szeg?, G.: Aufgaben und Lehrs?tze aus der Analysis I. Berlin Heidelberg New York: Springer 1954 |
| [14] | Poupard, Y.: Etude et denombrement paralleles des partitions non crois?es d’un cycle et des coupage d’un polygone convexe. Discrete Math.2, 279-288 (1972) · Zbl 0252.05005 · doi:10.1016/0012-365X(72)90008-8 |
| [15] | Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of M?bius functions. Z. Wahrscheinlichkeitstheor. Verw. Geb.2, 340-368 (1964) · Zbl 0121.02406 · doi:10.1007/BF00531932 |
| [16] | Rota, G.-C.: Finite operator calculus. New York: Academic Press 1975 |
| [17] | Shiryayev, A.N.: Probability. (Grad. Texts Math., vol. 95) Berlin Heidelberg New York: Springer 1984 |
| [18] | Simion, R.: Combinatorial statistics on non-crossing partitions. (Preprint 1991) · Zbl 0803.05003 |
| [19] | Simion, R., Ullman, D.: On the structure of the lattice of noncrossing partitions. Discrete Math.98, 193-206 (1991) · Zbl 0760.05004 · doi:10.1016/0012-365X(91)90376-D |
| [20] | Speicher, R.: A new example of ?Independence? and ?White Noise?. Probab. Theory Relat. Fields84, 141-159 (1990) · Zbl 0671.60109 · doi:10.1007/BF01197843 |
| [21] | Speicher, R.: The lattice of admissible partitions In: Accardi, L. (ed.) Quantum probability and related topics, vol. VIII. Singapore: World Scientific (to appear) · Zbl 0791.06010 |
| [22] | Speicher, R.: Free convolution and the random sum of matrices. (Publ., Res. Inst. Math. Sci., vol. 29) (to appear) · Zbl 0795.46048 |
| [23] | Voiculescu, D.: Symmetries of some reduced free productC*-algebras. In: Araki, H., Moore, C.C., Stratila, S., Voiculescu, D. (eds.) Operator algebras and their connection with topology and ergodic theory (Lect. Notes Math., vol. 1132, pp. 556-588) Berlin Heidelberg New York: Springer 1985 |
| [24] | Voiculescu, D.: Addition of certain non-commuting random variables. J. Funct. Anal.66, 323-346 (1986) · Zbl 0651.46063 · doi:10.1016/0022-1236(86)90062-5 |
| [25] | Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math.104, 201-220 (1991) · Zbl 0736.60007 · doi:10.1007/BF01245072 |
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