The Stirling numbers of the second kind, or Stirling partition numbers,
describe the number of ways a set with *n* elements can be partitioned into
*k* disjoint, non-empty subsets. Common notations are *S*(*n*, *k*),
, and
,
where the first is by far the easiest to type.

For example, the set {A, B, C} can be partitioned into one subset (that is, not partitioned at all)
in the following way, which means *S*(3, 1) = 1:

- {A, B, C}

into two subsets in the following ways (making *S*(3, 2) = 3):

- {A, B} {C}
- {A, C} {B}
- {A} {B, C}

and into three subsets in the following way (*S*(3, 3) = 1):

- {A} {B} {C}

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.

S(3, 1): | |

S(3, 2): | |

S(3, 3): |

S(4, 1): | |

S(4, 2): | |

S(4, 3): | |

S(4, 4): |

S(5, 1): | |

S(5, 2): | |

S(5, 3): | |

S(5, 4): | |

S(5, 5): |

You’ll notice that *S*(*n*, 1) is always 1: there’s only one way to partition
a set into one subset, which is to leave it alone.

You’ll also notice that *S*(*n*, *n*) is also always 1:
there’s only one way to partition a set of length *n* into *n* subsets,
each of length 1.

The Stirling numbers of the second kind also count numbers of ways to
place nonattacking rooks on a triangular checkerboard. (Compare with the *S*(4, *k*) diagrams.)

(Does the game of checkers use rooks?)

The numbers can be computed recursively using the formula:

*S*(*n*, *k*) = *S*(*n*−1, *k*−1) +
*S*(*n*−1, *k*)

The idea is that you can build up *S*(*n*, *k*) by:

- Taking each of the
*S*(*n*−1,*k*−1) partitions and adding singleton {*n*} - Taking each of the
*S*(*n*−1,*k*) partitions and appending element*n*to each of the*k*subsets in turn

For example, the following shows how to compute *S*(4, 3) by appending singleton {*e*}—represented
by the orange dot at the bottom of each square—to each of
the three *S*(3, 2) partitions, plus appending element *e* to each of the three
*S*(3, 3) subsets:

The sums of the Stirling numbers of the second kind,

are called the Bell numbers.

About the checkerboard interpretation, see M. B. Wells, Elements of Combinatorial Computing, Oxford: Pergamon Press, 1971 p. 185.

Designed and rendered using *Mathematica* 3.0 and 7.0.

© 1996–2023 by Robert Dickau.

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