Bell number

In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.

The Bell numbers are denoted ${\displaystyle B_{n}}$, where ${\displaystyle n}$ is an integer greater than or equal to zero. Starting with ${\displaystyle B_{0}=B_{1}=1}$, the first few Bell numbers are

${\displaystyle 1,1,2,5,15,52,203,877,4140,\dots }$ (sequence A000110 in the OEIS).

The Bell number ${\displaystyle B_{n}}$ counts the different ways to partition a set that has exactly ${\displaystyle n}$ elements, or equivalently, the equivalence relations on it. ${\displaystyle B_{n}}$ also counts the different rhyme schemes for ${\displaystyle n}$-line poems.^[1]

As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, ${\displaystyle B_{n}}$ is the ${\displaystyle n}$-th moment of a Poisson distribution with mean 1.