Definition:Bell Number

Definition

The Bell Numbers $B_n$ are a sequence of natural numbers defined as the number of ways a set with $n$ elements can be partitioned.

Sequence of Bell Numbers

The sequence of Bell numbers begins:

$(1), 1, 2, 5, 15, 52, 203, 877, 4140, 21 \, 147, \ldots$

where the first $(1)$ is the zeroth Bell number $B_0$.

Examples

Example: $B_2$

From the definition of Bell number, $B_2$ is the number of ways a set with $2$ elements can be partitioned.

$\set {a, b}$

$\set {\set a, \set b}$

$B_2 = 2$

Example: $B_3$

From the definition of Bell number, $B_3$ is the number of ways a set with $3$ elements can be partitioned.

$\set {a, b, c}$

$\set {\set a, \set {b, c} }$

$\set {\set b, \set {a, c} }$

$\set {\set c, \set {a, b} }$

$\set {\set a, \set b, \set c}$

$B_3 = 5$

Also see

• Results about Bell numbers can be found here.

Source of Name

This entry was named for Eric Temple Bell.