Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalised weights $W=(W_j)$ and $V=(V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,\ldots,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $V_{n,k}\,W_{|A_1|}\cdots W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $\Pi_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights $W$ must be of a very special form depending on a single parameter $\alpha\in[-\infty,1]$. The case $\alpha=1$ is trivial, and for each value of $\alpha\ne 1$ the set of possible $V$-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for $-\infty\le\alpha<0$ and continuous for $0\le\alpha<1$. For $\alpha\le 0$ the extremes correspond to the members of the Ewens–Pitman family of random partitions indexed by $(\alpha,\theta)$, while for $0<\alpha<1$ the extremes are obtained by conditioning an $(\alpha,\theta)$-partition on the asymptotics of the number of blocks of $\Pi_n$ as $n$ tends to infinity.