bell numbers

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

B_0=1,\quad B_1=1,\quad B_2=2,\quad B_3=5,\quad B_4=15,\quad B_5=52,\quad B_6=203,\quad\dots

In general, B_n 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 pairwise disjoint 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:

B_{n+1}=\sum_{k=0}^{n}{{n \choose k}B_k}.

They also satisfy "Dobinski's formula":

B_n=\frac{1}{e}\sum_{k=0}^\infty \frac{k^n}{k!}=

the nth moment of a Poisson distribution with expected value 1.

And they satisfy "Touchard's congruence": If p is any prime number then

B_{p+n}\equiv B_n+B_{n+1}\ (\operatorname{mod}\ p).

Each Bell number is a sum of "Stirling numbers of the second kind"

B_n=\sum_{k=1}^n S(n,k).

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. The exponential generating function of the Bell numbers is

\sum_{n=0}^\infty \frac{B_n}{n!} x^n = e^{e^x-1}.

Bell Numbers

See also