Stirling numbers of the second kind on Multiset 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?
There is no known formulation for a general multiset. However, a paper at JIS tackles the case where the element 1 occurs multiple times.
By definition, a set only has distinct elements.  If elements are duplicated, then yes, the set in question would be a multiset.