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Fragmenting random permutations

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Goldschmidt, C. (Christina), Martin, James B. and Spanò, Dario. (2008) Fragmenting random permutations. Electronic communications in probability, Vol.13 (No.44). pp. 461-474. ISSN 1083-589X

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Abstract

Problem 1.5.7 from Pitman's Saint-Flour lecture notes [11]: Does there exist for each n a fragmentation process (Pi(n,k); 1 <= k <= n) such that Pi(n,k) is distributed like the partition generated by cycles of a uniform random permutation of {1, 2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Permutations
Journal or Publication Title:	Electronic communications in probability
Publisher:	University of Washington. Dept. of Mathematics
ISSN:	1083-589X
Date:	14 August 2008
Volume:	Vol.13
Number:	No.44
Number of Pages:	14
Page Range:	pp. 461-474
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/D065755/1 (EPSRC), GR/T21783/01 (EPSRC)
References:	[1] N. Berestycki and J. Pitman. Gibbs distributions for random partitions generated by a fragmentation process. J. Stat. Phys., 127(2):381–418, 2007. [2] B. Efron. Increasing properties of P´olya frequency functions. Ann. Math. Statist., 36:272– 279, 1965. [3] P. Elias, A. Feinstein, and C. Shannon. A note on the maximum flow through a network. Institute of Radio Engineers, Transactions on Information Theory, IT-2:117–119, 1956. [4] L. R. Ford, Jr. and D. R. Fulkerson. Maximal flow through a network. Canad. J. Math., 8:399–404, 1956. [5] A. Gnedin and J. Pitman. Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. (N. Y.), 138(3):5677–5685, 2006. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102. [6] P. Hall. On representatives of subsets. J. London Math. Soc., 10:26–30, 1935. [7] S. G. Hoggar. Chromatic polynomials and logarithmic concavity. J. Combinatorial Theory Ser. B, 16, 1974. [8] T. Kamae, U. Krengel, and G. L. O’Brien. Stochastic inequalities on partially ordered spaces. Ann. Probab., 5(6):899–912, 1977. [9] J. Pitman. Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2006. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. [10] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25(2):855–900, 1997. [11] B. E. Sagan. Inductive and injective proofs of log concavity results. Discrete Math., 68(2- 3):281–292, 1988.
URI:	http://wrap.warwick.ac.uk/id/eprint/29515