the smallest integer of the block and $\pi _{i}=\varnothing $ otherwise. We let $\mathbb{O}=\{\{1\},\{2\},\ldots \}$ be the partition of $\mathbb{N}$ into singletons. If $B\subset
\mathbb{N}$ and $\pi \in \mathcal{P}_{\infty }$ we let $\pi \cap B$ (or $\pi
|_{B}$) be the restriction of $\pi $ to $B$, i.e. the partition of $B$ whose collection of blocks is $\{\pi _{i}\cap B,i\geq 1\}$. If $\pi \in
\mathcal{P}_{\infty }$ and $B\in \pi $ is a block of $\pi $, we let $$|B|=\lim_{n\rightarrow \infty }\frac{\#(B\cap \lbrack n])}{n}$$ be the asymptotic frequency of the block $B$, whenever it exists. A random variable $\pi $ with values in $\mathcal{P}_{\infty }$ is called *exchangeable* if its law is invariant by the natural action of permutations of $\mathbb{N}$ on $\mathcal{P}_{\infty }$. By a theorem of Kingman [@kingman78; @aldous85], all the blocks of such random partitions admit asymptotic frequencies a.s. For $\pi $ whose blocks have asymptotic frequencies, we let $|\pi |\in S$ be the decreasing sequence of these frequencies. Kingman’s theorem more precisely says that the law of any exchangeable random partition $\pi $ is a (random) “paintbox process”, a term we now explain. Take $\mathbf{s}\in S$ (the paintbox) and consider a sequence $U_{1},U_{2},\ldots $ of i.i.d. variables in 