Set Partitions & Stirling Numbers Calculator

S(n, k) is the number of ways to partition a set of n distinct objects into exactly k non-empty subsets, where the subsets themselves are not labeled or ordered. It satisfies the recurrence S(n, k) = k · S(n−1, k) + S(n−1, k−1), with S(n, n) = 1 and S(n, 0) = 0 for n ≥ 1.

If the groups are interchangeable (e.g. identical boxes), use S(n, k). If the groups are distinguishable (e.g. numbered boxes, or people receiving them), multiply by k!, because each partition can be assigned to the labeled groups in k! ways. That labeled count, k!·S(n, k), also equals the number of ways to map n objects onto k groups so that none is empty (surjections).

The Bell number B(n) is the total number of ways to partition n objects into any number of non-empty groups, from 1 up to n. It is the sum of S(n, k) over all k, so B(n) = S(n, 1) + S(n, 2) + … + S(n, n).

Combinations count how many ways to choose r items from n, ignoring order — a single subset. Set partitions count how many ways to split the whole set into several non-empty groups at once. For choosing items, use the permutations and combinations calculator instead.