  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 $\operatorname
{PD}(\alpha,\theta)$ that take values in this space. We introduce families of
random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and
$\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if
the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$
distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$
distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$.
This result may be proved using a subordinator representation and it provides a
companion set of relations to those of Pitman between $\operatorname
{PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated
application of the $\mathrm {Frag}_{\alpha}$ 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.
