Electronic Communications in Probability

Fragmenting random permutations

Christina Goldschmidt, James Martin, and Dario Spano

Full-text: Open access


Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each $n$ a fragmentation process $(\Pi_{n,k}, 1 \leq k \leq n)$ such that $\Pi_{n,k}$ is distributed like the partition generated by cycles of a uniform random permutation of $\{1,2,\ldots,n\}$ conditioned to have $k$ cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.

Article information

Electron. Commun. Probab., Volume 13 (2008), paper no. 44, 461-474.

Accepted: 14 August 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05A18: Partitions of sets

Fragmentation process random permutation Gibbs partition Chinese restaurant process

This work is licensed under aCreative Commons Attribution 3.0 License.


Goldschmidt, Christina; Martin, James; Spano, Dario. Fragmenting random permutations. Electron. Commun. Probab. 13 (2008), paper no. 44, 461--474. doi:10.1214/ECP.v13-1402. https://projecteuclid.org/euclid.ecp/1465233471

Export citation


  • Berestycki, Nathanaël; Pitman, Jim. Gibbs distributions for random partitions generated by a fragmentation process. J. Stat. Phys. 127 (2007), no. 2, 381–418.
  • Elias, P.; Feinstein, A.; Shannon, C.. A note on the maximum flow through a network. Institute of Radio Engineers, Transactions on Information Theory IT-2 (1956), 117–119.
  • Ford, L. R., Jr.; Fulkerson, D. R. Maximal flow through a network. Canad. J. Math. 8 (1956), 399–404.
  • Gnedin, A.; Pitman, J. Exchangeable Gibbs partitions and Stirling triangles. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 325 (2005), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 12, 83–102, 244–245; translation in J. Math. Sci. (N. Y.) 138 (2006), no. 3, 5674–5685
  • Gnedin, Alexander; Pitman, Jim. Poisson representation of a Ewens fragmentation process. Combin. Probab. Comput. 16 (2007), no. 6, 819–827.
  • Granovsky, Boris; Erlihson, Michael. On time dynamics of coagulation-fragmentation processes. arXiv:0711.0503v2[math.PR] (2007).
  • Hall, P.. On representatives of subsets. J. London Math. Soc. 10 (1935), 26–30.
  • Hoggar, S. G. Chromatic polynomials and logarithmic concavity. J. Combinatorial Theory Ser. B 16 (1974), 248–254.
  • Kamae, T.; Krengel, U.; O'Brien, G. L. Stochastic inequalities on partially ordered spaces. Ann. Probability 5 (1977), no. 6, 899–912.
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855–900.
  • Sagan, Bruce E. Inductive and injective proofs of log concavity results. Discrete Math. 68 (1988), no. 2-3, 281–292.