Stirling numbers of the second kind

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The Stirling numbers of the second kind describe the number of ways a set with n elements can be partitioned into k disjoint, 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 Mathematica 3.0 for the Apple Macintosh.

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