Geodesic
Random partitions of sets with marked subsets
Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 485-498 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[2] N. G. de Brein, Asimptoticheskie metody v analize, IL, Moskva, 1961

[3] L. Moser, M. Wyman, “An asymptotic formula for the Bell numbers”, Trans. Roy. Soc. Canada, 49 (1955), 49–54 | MR | Zbl

[4] L. H. Harper, “Stiriling behavior is asymptotically normal”, Ann. Math. Statist., 38:2 (1967), 410–414 | DOI | MR | Zbl

[5] A. Renyi, “Uj modzerek és eredmemy a kombinatorikus analizisben, I”, Magyar. Tud. Akad. Mat. Fiz., Tud. Oszt. Közl., 16:1 (1966), 77–105 | MR | Zbl

[6] I. H. Curtiss, “A note on the theory of moment generating functions”, Ann. Math. Statist., 13:3 (1942), 430–433 | DOI | MR | Zbl

[7] V. N. Sachkov, Perechislitelnye metody kombinatornogo analiza, Voprosy kombinatornoi matematiki. Izd. nauchnogo soveta po kompleksnoi probleme kibernetika, AN SSSR, Moskva, 1973 | Zbl

[8] V. N. Sachkov, “Raspredelenie chisla razlichnykh elementov simmetrichnogo bazisa v sluchainoi $mA$-vyborke”, Teoriya veroyatnostei, XVI:3 (1971), 504–513