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The Stirling numbers of the second kind describe the number of ways a set with

nelements can be partitioned intokdisjoint, non-empty subsets.For example, the set {1, 2, 3} can be partitioned into three subsets in the following way --

{{1}, {2}, {3}}, into two subsets in the following ways --

{{1, 2}, {3}} {{1, 3}, {2}} {{1}, {2, 3}}, and into one subset in the following way --

{{1, 2, 3}}. The numbers can be computed recursively using this formula:

. 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.

`StirlingS2[3, k]`

:

`StirlingS2[4, k]`

:

`StirlingS2[5, k]`

:

`StirlingS2[6, k]`

:

The sums of the Stirling numbers of the second kind,

, are called the

Bell numbers.Designed and rendered using

Mathematica3.0 for the Apple Macintosh.

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