Random mappings with a given number of cyclical points

Jennie C. Hansen, Jerzy Jaworski

Research output: Contribution to journalArticle

Abstract

In this paper we consider a random mapping, T^n, of the finite set {1, 2,..., n} into itself for which the digraph representation Gn is constructed by: (1) selecting a random number, L^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 a random permutation to the selected cyclic vertices. We investigate k^n, the size of a 'typical' component of Gn, and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution of k^n conditioned on L^n = m(n). As an application of our results, we show in Section 3 that provided L^n is of order much larger than vn, then the joint distribution of the normalized order statistics of the component sizes of Gn converges to the Poisson-Dirichlet(1) distribution as n ? 8. Other applications and generalizations are also discussed in Section 3.

Original languageEnglish
Pages (from-to)341-359
Number of pages19
JournalArs Combinatoria
Volume94
Publication statusPublished - Jan 2010

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