On the number of allelic types for samples taken from exchangeable coalescents with mutation

Let $K_n$ denote the number of types of a sample of size $n$ taken from an exchangeable coalescent process ($\mathrm{\Xi }$-coalescent) with mutation. A distributional recursion for the sequence $(K_n{)}_{n\in \mathbb{N}}$ is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure $\mathrm{\Xi }$ on the infinite simplex $\mathrm{\Delta }$ does not have mass at zero and satisfies ${\int }_{\mathrm{\Delta }}\mathrm{\mid }x\mathrm{\mid }\mathrm{\Xi }(dx)\mathrm{/}(x,x)<\mathrm{\infty }$, where $|x|:=\sum_{i=1}^\infty x_i$and$(x,x):=\sum_{i=1}^\infty x_i^2$for$x=(x_1,x_2,…)\in\Delta$,then$K_n/n$convergesweaklyas$n\to\infty$toalimitingvariable$K$ which is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures $\mathrm{\Xi }$ satisfying ${\int }_{\mathrm{\Delta }}\mathrm{\Xi }(dx)\mathrm{/}(x,x)<\mathrm{\infty }$ we characterize the distribution of $K$ via a fixed-point equation.