The Multinomial Distribution on Rooted Labeled Forests

For a probability distribution p s ; 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 p s ; s2 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.