  Gibbs-type exchangeable random partitions, which is a class of multiplicative
measures on the set of positive integer partitions, appear in various contexts,
including Bayesian statistics, random combinatorial structures, and stochastic
models of diversity in various phenomena. Some distributional results on
ordered sizes in the Gibbs partition are established by introducing associated
partial Bell polynomials and analysis of the generating functions. The
combinatorial approach is applied to derive explicit results on asymptotic
behavior of the extreme sizes in the Gibbs partition. Especially, Ewens-Pitman
partition, which is the sample from the Poisson-Dirichlet process and has been
discussed from rather model-specific viewpoints, and a random partition which
was recently introduced by Gnedin, are discussed in the details. As
by-products, some formulas for the associated partial Bell polynomials are
presented.
