Bell numbers
The
Bell numbers, named in honor of
Eric Temple Bell, are a sequence of integers arising in
combinatorics that begins thus:
-
In general,
Bn is the number of
partitions of a set of size
n. (
B0 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:
-
They also satisfy "Dobinski's formula":
-
And they satisfy "Touchard's congruence": If
p is any
prime number then
Each Bell number is a sum of "Stirling numbers of the second kind"
-
The Stirling number
S(
n,
k) is the number of ways to partition a set of cardinality
n into exactly
k nonempty subsets.
The nth Bell number is also the sum of the coefficients in the polynomial that expresses the nth 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.
