Editing Bell number (section)

{{Short description|Count of the possible partitions of a set}}

{{distinguish|Pell number}}

In [[combinatorics|combinatorial mathematics]], the '''Bell numbers''' count the possible [[partition of a set|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 <math>B_n</math>, where <math>n</math> is an [[integer]] greater than or equal to [[zero]]. Starting with <math>B_0 = B_1 = 1</math>, the first few Bell numbers are
:<math>1, 1, 2, 5, 15, 52, 203, 877, 4140, \dots</math> {{OEIS|id=A000110}}.
The Bell number <math>B_n</math> counts the different ways to partition a set that has exactly <math>n</math> elements, or equivalently, the [[equivalence relation]]s on it. <math>B_n</math> also counts the different [[rhyme scheme]]s for <math> n </math>-line poems.{{sfn|Gardner|1978}}

As well as appearing in counting problems, these numbers have a different interpretation, as [[moment (mathematics)|moments]] of [[probability distribution]]s. In particular, <math>B_n</math> is the <math> n </math>-th moment of a [[Poisson distribution]] with [[mean]] 1.