Bell number: Difference between revisions

The Bell numbers, named in honor of Eric Temple Bell, are a sequence of integers 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 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":

$B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}={\rm {\ the\ }}n{\rm {th\ moment\ of\ a\ Poisson\ distribution\ with\ expected\ value\ 1}}.$

[As it stands, this article is somewhat stubby. I may return to it later; as may others.]