$\mathbb{N}\cup \{0\}$ (the colors) with $P(U_{1}=j)=s_{j}$ for $j\geq 1$ and $P(U_{1}=0)=1-\sum_{k}s_{k}$. Define a partition $\pi $ on $\mathbb{N}$ by saying that $i\neq j$ are in the same block if and only if $U_{i}=U_{j}\neq 0$ (i.e. $i$ and $j$ have the same color, where $0$ is considered as colorless). Call $\rho _{\mathbf{s}}(\mathrm{d}\pi )$ its law, the *$\mathbf{s}$-paintbox* law. Kingman’s theorem says that the law of any random partition is a mixing of paintboxes, i.e. it has the form $\int_{\mathbf{s}\in S}m(\mathrm{d}{\mathbf{s}})\rho _{\mathbf{s}}(\mathrm{d}\pi )$ for some probability measure $m$ on $S$. A useful consequence is that the block of an exchangeable partition $\pi $ containing $1$, or some prescribed integer $i$, is a *size-biased pick* from the blocks of $\pi $, i.e. the probability it equals a non-singleton block $\pi _{j}$ conditionally on $(|\pi _{j}|,j\geq 1)$ equals $|\pi _{j}|$. Similarly,

\[usefull\] Let $\pi $ be an exchangeable random partition which is a.s. different from the trivial partition $\mathbb{O}$, and $B$ an infinite subset of $\mathbb{N}$. For any $i\in \mathbb{N}$, let $$\widetilde{i}=\inf \{j\geq i:j\in B\mbox{ and }\{j\}\notin \pi \},$$ then $\widetilde{i}<\infty $ a.s. and the block $\widetilde{\pi}$ of $\pi $ containing $\widetilde{i}$ is a size-biased pick among the non-singleton blocks of $\pi $, i.e. 