Bell numbers

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

<math>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</math>

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 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:

<math>B_{n+1}=\sum_{k=0}^{n}{{n \choose k}B_k}.</math>

They also satisfy "Dobinski's formula":

<math>B_n=\frac{1}{e}\sum_{k=0}^\infty \frac{k^n}{k!}={\rm\ the\ }n{\rm th\ moment\ of\ a\ Poisson\ distribution\ with\ expected\ value\ 1}.</math>

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

<math>B_{p+n}\equiv B_n+B_{n+1}\ (\operatorname{mod}\ p).</math>

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

<math>B_n=\sum_{k=1}^n S(n,k).</math>

The Stirling number S(n, k) is the number of ways to partition a set of cardinality n into exactly k nonempty subsets.