  Gibbs partitions of the integers generated by stable subordinators of index
$\alpha\in(0,1)$ form remarkable classes of random partitions where in
principle much is known about their properties, including practically
effortless obtainment of otherwise complex asymptotic results potentially
relevant to applications in general combinatorial stochastic processes, random
tree/graph growth models and Bayesian statistics. This class includes the
well-known models based on the two-parameter Poisson-Dirichlet distribution
which forms the bulk of explicit applications. This work continues efforts to
provide interpretations for a larger classes of Gibbs partitions by embedding
important operations within this framework. Here we address the formidable
problem of extending the dual, infinite-block, coagulation/fragmentation
results of Jim Pitman (1999, Annals of Probability), where in terms of
coagulation they are based on independent two-parameter Poisson-Dirichlet
distributions, to all such Gibbs (stable Poisson-Kingman) models. Our results
create nested families of Gibbs partitions, and corresponding mass partitions,
over any $0<\beta<\alpha<1.$ We primarily focus on the fragmentation
operations, which remain independent in this setting, and corresponding
remarkable calculations for Gibbs partitions derived from that operation. We
also present definitive results for the dual coagulation operations, now based
on our construction of dependent processes, and demonstrate its relatively
simple application in terms of Mittag-Leffler and generalized gamma models. The
latter demonstrates another approach to recover the duality results in Pitman
(1999).

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