# Bell number diagrams

The Bell numbers

The

Here are some diagrams representing the different ways the sets can be partitioned: a line connects elements in the same subset, and a point represents a singleton subset.

{1, 2, 3}, 5 partitions:

{1, 2, 3, 4}, 15 partitions:

{1, 2, 3, 4, 5}, 52 partitions:

{1, 2, 3, 4, 5, 6}, 203 partitions:

{1, 2, 3, 4, 5, 6, 7}, 877 partitions:

(The GIF image is about 125K, so let me know if you're dying to see it.)

Copyright © 1996 Robert M. Dickau

(1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...)

describe the number of ways a set with *n*elements can be partitioned into disjoint, non-empty subsets.For example, the set {1, 2, 3} can be partitioned in the following ways:

- {{1}, {2}, {3}}
- {{1, 2}, {3}}
- {{1, 3}, {2}}
- {{1}, {2, 3}}
- {{1, 2, 3}}.

The

*n*th Bell number can be computed using the formula$\sum_{k=0}^n S_n^{(k)}$

where $S_n^{(k)}$ represents the Stirling numbers of the second kind.
Here are some diagrams representing the different ways the sets can be partitioned: a line connects elements in the same subset, and a point represents a singleton subset.

{1, 2, 3}, 5 partitions:

{1, 2, 3, 4}, 15 partitions:

{1, 2, 3, 4, 5}, 52 partitions:

{1, 2, 3, 4, 5, 6}, 203 partitions:

{1, 2, 3, 4, 5, 6, 7}, 877 partitions:

(The GIF image is about 125K, so let me know if you're dying to see it.)

Copyright © 1996 Robert M. Dickau