  We prove a long-standing conjecture which characterises the Ewens-Pitman
two-parameter family of exchangeable random partitions, plus a short list of
limit and exceptional cases, by the following property: for each $n = 2,3,
>...$, if one of $n$ individuals is chosen uniformly at random, independently
of the random partition $\pi_n$ of these individuals into various types, and
all individuals of the same type as the chosen individual are deleted, then for
each $r > 0$, given that $r$ individuals remain, these individuals are
partitioned according to $\pi_r'$ for some sequence of random partitions
$(\pi_r')$ that does not depend on $n$. An analogous result characterizes the
associated Poisson-Dirichlet family of random discrete distributions by an
independence property related to random deletion of a frequency chosen by a
size-biased pick. We also survey the regenerative properties of members of the
two-parameter family, and settle a question regarding the explicit arrangement
of intervals with lengths given by the terms of the Poisson-Dirichlet random
sequence into the interval partition induced by the range of a neutral-to-the
right process.
