@MISC{Pitman97themultinomial, author = {Jim Pitman}, title = {The Multinomial Distribution on Rooted Labeled Forests}, year = {1997} }

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Abstract

For a probability distribution (p s ; s 2 S) on a finite 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 ; 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. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS97-03961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...