Random mappings with a given number of cyclical points

In this paper we consider a random mapping, T̂n, of the finite set {1, 2, ..., n} into itself for which the digraph representation Ĝn is constructed by: (1) selecting a random number, L̂n, 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 k̂n, the size of a ‘typical’ component of Ĝn, and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution of k̂n conditioned on L̂n = m(n). As an application of our results, we show in Section 3 that provided L̂n is of order much larger than √ n, then the joint distribution of the normalized order statistics of the component sizes of Ĝn converges to the Poisson-Dirichlet(1) distribution as n → ∞. Other applications and generalizations are also discussed in Section 3.