Geodesic
Random partitions of sets with marked subsets
Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 485-498

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We are given a uniform probability distribution on the partitions of a set $X$ of $m$ elements. For each realization of the random partition we define a random process of drawing marks: every subset of cardinality $k$ receives a mark with probability $p_k$. We find expressions for exact distributions of the number of marked subsets of cardinality $l$, the overall number of marked subsets and the number of elements in them. For certain concrete values $p_k=p_k(m)$, $k=1,2,\dots,$ we obtain the limit distributions of these random variables as $m\to\infty$. Bibliography: 8 titles. 
@article{SM_1973_21_3_a9,
     author = {V. N. Sachkov},
     title = {Random partitions of sets with marked subsets},
     journal = {Sbornik. Mathematics},
     pages = {485--498},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1973_21_3_a9/}
}
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V. N. Sachkov. Random partitions of sets with marked subsets. Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 485-498. http://geodesic.mathdoc.fr/item/SM_1973_21_3_a9/