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 (α,θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag_α and Coag_α,θ, respectively, with the following property: if the input to Frag_α has PD (α,θ) distribution, then the output has PD (α,θ+1) distribution, while the reverse is true for Coag_α,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD (α,θ) and PD (αβ,θ). Repeated application of the Frag_α 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.