Coagulation.fragmentation duality, Poisson.Dirichlet distributions and random recursive trees

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson.Dirichlet distributions PD.(.,.) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag. and Coag.,., respectively, with the following property: if the input to Frag. has PD.(.,.) distribution, then the output has PD.(.,.+1) distribution, while the reverse is true for Coag.,.. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD.(.,.) and PD.(..,.). Repeated application of the Frag. operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation.fragmentation duality.