  We introduce a new random mapping model, $T_n^{\hat D}$, which maps the set
$\{1,2,...,n\}$ into itself.The random mapping $T_n^{\hat D}$ is constructed
using a collection of exchangeable random variables $\hat{D}_1, ....,\hat{D}_n$
which satisfy $\sum_{i=1}^n\hat{D}_i=n$. In the random digraph,$G_n^{\hat D}$,
which represents the mapping $T_n^{\hat D}$, the in-degree sequence for the
vertices is given by the variables $\hat{D}_1, \hat{D}_2, ..., \hat{D}_n$, and,
in some sense,$G_n^{\hat D}$ 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 $\hat{D}_1, \hat{D}_2, ..., \hat{D}_n$. We also consider
two special examples of $T_n^{\hat D}$ 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.
