Gibbs distributions for random partitions generated by a fragmentation process

In this paper we study random partitions of {1, . . . , n} where every cluster of size j can be in any of wj possible internal states. The Gibbs (n, k, w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n, k, w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences wj , the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate Ki,j = a + b(i + j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.