Random mappings with exchangeable in-degrees

In this article we introduce a new random mapping model, T-n((D) over cap) , which maps the set {1, 2, n} into itself. The random mapping T-n((D) over cap) is constructed using a collection of exchangeable random variablesb (D) over cap (1),...,(D) over cap (n) which satisfy Sigma(n)(i=1) (D) over cap (i) = n. In the random digraph, G(n)((D) over cap), which represents the mapping T-n((D) over cap), the in-degree sequence for the vertices is given by the variables (D) over cap (1), (D) over cap (2),...,(D) over cap (n) and, in some sense, G(n)((D) over cap) can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of (D) over cap (1), (D) over cap (2),..., (D) over cap (n). We also consider two special examples of T-n((D) over cap) which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. (C) 2007 Wiley Periodicals, Inc.