Random partitions of sets with marked subsets

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 $m\to\infty$.
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