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Gibbs partitions of the integers generated by stable subordinators of index$\alpha\in(0,1)$ form remarkable classes of random partitions where inprinciple much is known about their properties, including practicallyeffortless obtainment of otherwise complex asymptotic results potentiallyrelevant to applications in general combinatorial stochastic processes, randomtree/graph growth models and Bayesian statistics. This class includes thewell-known models based on the two-parameter Poisson-Dirichlet distributionwhich forms the bulk of explicit applications. This work continues efforts toprovide interpretations for a larger classes of Gibbs partitions by embeddingimportant operations within this framework. Here we address the formidableproblem of extending the dual, infinite-block, coagulation/fragmentationresults of Jim Pitman (1999, Annals of Probability), where in terms ofcoagulation they are based on independent two-parameter Poisson-Dirichletdistributions, to all such Gibbs (stable Poisson-Kingman) models. Our resultscreate nested families of Gibbs partitions, and corresponding mass partitions,over any $0<\beta<\alpha<1.$ We primarily focus on the fragmentationoperations, which remain independent in this setting, and correspondingremarkable calculations for Gibbs partitions derived from that operation. Wealso present definitive results for the dual coagulation operations, now basedon our construction of dependent processes, and demonstrate its relativelysimple application in terms of Mittag-Leffler and generalized gamma models. Thelatter demonstrates another approach to recover the duality results in Pitman(1999).

Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators