Partitions of multisets

A multiset is a set with repeated elements. There are four distinct partition numbers to consider, unlike the classical set partition case which involves only Stirling numbers of the second kind. Using inclusion-exclusion, we obtain generating functions when each element appears exactly r = 1, 2 or 3 times. The case r = 1 is classical and r = 2 was studied by Comtet and Baróti using other methods. Our approach also leads to asymptotic formulae for the total number of partitions of multisets in which the repetition of elements is bounded. Another approach to multiset enumeration, using de Brujin's theorem for group reduced distributions, is described.