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Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of $n$ elements into $k$ nonempty subsets. What if there were duplicate elements in the set? That is, the set is a multiset?

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There is no known formulation for a general multiset. However, a paper at JIS tackles the case where the element 1 occurs multiple times.

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Here are two links to get you started:

  1. Eulerian numbers of the second kind may be helpful (for counting ascents, descents, etc., though i think)

  2. Additionally some more useful information may be found in Stanley's book

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You divide the Stirling number by the possible permutations of identical elements as this is just changing order within the stars and bars example.

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By definition, a set only has distinct elements. If elements are duplicated, then yes, the set in question would be a multiset.

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