The

**Bell numbers**, named in honor of Eric Temple Bell, are a sequence of integers arising in combinatorics that begins thus:

*B*

_{n}is the number of partitions of a set of size

*n*. (

*B*

_{0}is 1 because there is exactly one partition of the empty set. A partition of a set

*S*is by definition a set of nonempty sets whose union is

*S*. Every member of the empty set is a nonempty set (that is vacuously true), and their union is the empty set. Therefore, the empty set is the only partition of itself.)

The Bell numbers satisfy this recursion formula:

*p*is any prime number then

*S*(

*n*,

*k*) is the number of ways to partition a set of cardinality

*n*into exactly

*k*nonempty subsets.

The *n*th Bell number is also the sum of the coefficients in the polynomial that expresses the *n*th moment of any probability distribution as a function of the first *n* cumulants; this way of enumerating partitions is not as coarse as that given by the Stirling numbers.