Bell numbers

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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. 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:

{{a}, {b}, {c}}

{{a}, {b, c}}

{{b}, {a, c}}

{{c}, {a, b}}

B₀ is 1 because there is exactly one partition of the empty set. 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 n-th 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 n-th 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