  Random discrete distributions, say $F,$ known as species sampling models,
represent a rich class of models for classification and clustering, in Bayesian
statistics and machine learning. They also arise in various areas of
probability and its applications. Jim Pitman, within the species sampling
context, shows that mixed Poisson processes may be interpreted as the sample
size up till a given time or in terms of waiting times of appearance of
individuals to be classified. He notes connections to some recent work in the
Bayesian statistic/machine learning literature, with some more classical
results. We let $F:=F_{\alpha,\theta},$ be a Pitman-Yor process for $\alpha\in
(0,1),$ and $\theta>-\alpha,$ with $\alpha$-diversity equivalent in
distribution to $S^{-\alpha}_{\alpha,\theta},$ and let
$(N_{S_{\alpha,\theta}}(\lambda),\lambda\ge 0)$ denote a mixed Poisson process
with rate $S_{\alpha,\theta}.$ In this paper we derive explicit stick-breaking
representations of $F_{\alpha,\theta}$ given
$N_{S_{\alpha,\theta}}(\lambda)=m.$ More precisely, if $(P_{\ell})\sim
\mathrm{PD}(\alpha,\theta)$, denotes a ranked sequence following the two
parameter Poisson-Dirichlet distribution, we obtain explicit representations of
the sized biased permutation of $(P_{\ell})|N_{S_{\alpha,\theta}}(\lambda)=m.$
Due to distributional results we shall develop in a more general context, it
suffices to consider the stable case $F_{\alpha,0}|N_{S_{\alpha}}(\lambda)=m.$
Notably, it follows that $F_{\alpha,0}|N_{S_{\alpha}}(\lambda)=0,$ is
equivalent in distribution to the popular normalized generalized gamma process.
Hence, we obtain explicit stick-breaking representations for the generalized
gamma class as a special case.
