Regeneration in random combinatorial structures

Abstract

Kingman’s theory of partition structures relates, via a natural sampling procedure, finite partitions to hypothetical infinite populations. Explicit formulas for distributions of such partitions are rare, the most no- table exception being the Ewens sampling formula, and its two-parameter extension by Pitman. When one adds an extra structure to the partitions like a linear order on the set of blocks and regenerative properties, some representation theorems allow to get more precise information on the dis- tribution. In these notes we survey recent developments of the theory of regenerative partitions and compositions. In particular, we discuss connec- tion between ordered and unordered structures, regenerative properties of the Ewens-Pitman partitions, and asymptotics of the number of compo- nents.