Bell numbers

Definition

Let $n$ be a natural number. The Bell number $B_n$ is the number of partitions of an $n$-element set into nonempty blocks. Equivalently, it is the sum of Stirling numbers of the second kind: $$B_n = \sum_{k=0}^{n} S(n, k).$$

Remarks

The first values, for $n = 0, 1, \dots, 6$, are $1, 1, 2, 5, 15, 52, 203$. $B_0 = 1$, because the empty set has exactly one partition, the partition with no blocks. The values satisfy a recurrence with binomial coefficients.

Explanations

L1 human

The Bell number $B_n$ counts all ways to split $n$ labelled objects into groups, with any number of groups. For $\{1, 2, 3\}$ there are $5$ ways: $123$, $1|23$, $2|13$, $3|12$ and $1|2|3$.