Bell number

In combinatorial mathematics, the Bell numbers count the number of partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s.

Starting with B₀ = B₁ = 1, the first few Bell numbers are:

The nth of these numbers, B_n, counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Outside of mathematics, the same number also counts the number of different rhyme schemes for n-line poems.

As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, B_n is the nth moment of a Poisson distribution with mean 1.

What these numbers count

Set partitions

In general, B_n is the number of partitions of a set of size n. A partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B₃ = 5 because the 3-element set {a, b, c} can be partitioned in 5 distinct ways: