How to calculate the number of possible multiset partitions into N disjoint sets?

combinatorics algorithms prime-factorization set-partition proof-verification

Solution 1:

The number of partitions of a set of size $n$ is called the Bell number of $n$ and is denoted $B_n$. See e.g. https://en.m.wikipedia.org/wiki/Bell_number for more.

The notion can be generalised to multisets. See https://www.sciencedirect.com/science/article/pii/0012365X74900764

Also see Finding the amount partitions of a multiset

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