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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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    It would be good to add the link to the definition of Stirling numbers: http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind2011-01-09
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    The answer is likely to be somewhat complicated to express. For example, if the set consists of a single type of element, you get the partition numbers p_{n,k} : http://en.wikipedia.org/wiki/Partition_(number_theory)2011-01-09
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    I think [math] is a pretty useless tag in this site; I've deleted it and added [counting]2011-01-10

4 Answers 4

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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.