### 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 G_{n} 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 G_{n}, 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 G_{n} converges to the Poisson-Dirichlet(1) distribution as n ? 8. Other applications and generalizations are also discussed in Section 3.

Original language | English |
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Pages (from-to) | 341-359 |

Number of pages | 19 |

Journal | Ars Combinatoria |

Volume | 94 |

Publication status | Published - Jan 2010 |

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*Ars Combinatoria*,

*94*, 341-359.

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*Ars Combinatoria*, vol. 94, pp. 341-359.

**Random mappings with a given number of cyclical points.** / Hansen, Jennie C.; Jaworski, Jerzy.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Random mappings with a given number of cyclical points

AU - Hansen, Jennie C.

AU - Jaworski, Jerzy

N1 - The research in this paper was supported by the Marie Curie Intra-European Fellowship No. 501863 (RANDIGRAPH) within the 6th European Community Framework Programme.

PY - 2010/1

Y1 - 2010/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77949525176&partnerID=8YFLogxK

M3 - Article

VL - 94

SP - 341

EP - 359

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -