Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests, and subsets

Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is de ned by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees.