Random mappings with Ewens cycle structure
Authors:
- Jennie C. Hansen,
- Jerzy Jaworski
Abstract
In this paper we consider a random mapping, T̂
n,θ
, of the finite set {1,2,...,n} into itself for which the digraph representation Ĝ
n,θ
is constructed by: (1) selecting a random number, L̂
n
, of cyclic vertices, (2) constructing a uniform random forest of size n with the selected cyclic vertices as roots, and (3) forming 'cycles' of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter θ. We investigate k̂
n,θ
, the size of a 'typical' component of Ĝ
n,θ
, and we obtain the asymptotic distribution of k̂
n,θ
conditioned on L̂
n
= m(n). As an application of our results, we show in Section 3 that provided L̂
n
is of order much larger than √n, then the joint distribution of the normalized order statistics of the component sizes of Ĝ
n,θ
converges to the Poisson-Dirichlet(θ) distribution as n→∞.
- Record ID
- UAMd4c5c3cdd614490eba94a6a7d8ed23ab
- Author
- Journal series
- Ars Combinatoria, ISSN 0381-7032
- Issue year
- 2013
- Vol
- 112
- Pages
- 307-322
- ASJC Classification
- Language
- (en) English
- Score (nominal)
- 15
- Score source
- journalList
- Score
- Publication indicators
- = 1; = 1; : 2013 = 0.702; : 2013 (2 years) = 0.204 - 2013 (5 years) =0.257
- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMd4c5c3cdd614490eba94a6a7d8ed23ab/
- URN
urn:amu-prod:UAMd4c5c3cdd614490eba94a6a7d8ed23ab
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