Properties of the structure of random partitions of a finite set in parametric model

Some structural properties of random partitions of a finite $n$-element set are investigated when a parametric probabilistic measure on the set of all partitions is defined. Asymptotic behavior of several structural characteristics (as $n \to \infty$) is studied: sets of numbers of blocks with given sizes, numbers of small and large blocks, numbers of blocks with medium sizes, extremal and related characteristics of structure.