Electronic Communications in Probability

On the number of cycles in a random permutation

Kenneth Maples, Ashkan Nikeghbali, and Dirk Zeindler

Full-text: Open access

Abstract

We show that the number of cycles in a random permutation chosen according to generalized Ewens measure is normally distributed and compute asymptotic estimates for the mean and variance.

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 20, 13 pp.

Dates
Accepted: 27 May 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263153

Digital Object Identifier
doi:10.1214/ECP.v17-1934

Mathematical Reviews number (MathSciNet)
MR2943103

Zentralblatt MATH identifier
1243.60010

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Keywords
random permutation generalized Ewens measure total number of cycles central limit theorem large deviations

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Maples, Kenneth; Nikeghbali, Ashkan; Zeindler, Dirk. On the number of cycles in a random permutation. Electron. Commun. Probab. 17 (2012), paper no. 20, 13 pp. doi:10.1214/ECP.v17-1934. https://projecteuclid.org/euclid.ecp/1465263153


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References

  • Arratia, Richard; Barbour, Andrew; Tavaré, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. xii+363 pp. ISBN: 3-03719-000-0
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and Poisson-Dirichlet law of cycle lengths. Electron. J. Probab. 16 (2011), no. 41, 1173–1192.
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009), no. 2, 469–501.
  • Betz, Volker; Ueltschi, Daniel; Velenik, Yvan. Random permutations with cycle weights. Ann. Appl. Probab. 21 (2011), no. 1, 312–331.
  • Ercolani, Nicholas; Ueltschi, Daniel. Cycle structure of random permutations with cycle weights. Preprint 2011
  • Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5
  • Nikeghbali, Ashkan; Zeindler, Dirk. The generalized weighted probability measure on the symmetric group and the asymptotic behaviour of the cycles, To appear in Annales de L'Institut Poincaré, 2011.