The multinomial distribution on rooted labeled forests

For a probability distribution (ps; s 2 S) on a nite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a p-forest if given F has m edges the vector of out-degrees of vertices of F has a multinomial distribution with parameters m and (ps; s 2 S), and given also these out-degrees the distribution of F is uniform on all forests with the given out-degrees. The family of distributions of p-forests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries.