The number of ways of partitioning a set of elements into nonempty Sets (i.e., Blocks), also called a Stirling Set Number. For example, the Set can be partitioned into three Subsets in one way: ; into two Subsets in three ways: , , and ; and into one Subset in one way: .

The Stirling numbers of the second kind are denoted , , , or
, so the Stirling numbers of the second kind for three elements are

(1) | |||

(2) | |||

(3) |

Since a set of elements can only be partitioned in a single way into 1 or Subsets,

(4) |

The Stirling numbers of the second kind can be computed from the sum

(5) |

(6) |

(7) |

(8) |

The following diagrams (Dickau) illustrate the definition of the Stirling numbers of the second kind for and 4.

Stirling numbers of the second kind obey the Recurrence Relations

(9) |

An identity involving Stirling numbers of the second kind is

(10) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Stirling Numbers of the Second Kind.'' §24.1.4 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 824-825, 1972.

Comtet, L. *Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed.* Boston, MA: Reidel, 1974.

Conway, J. H. and Guy, R. K. In *The Book of Numbers.* New York: Springer-Verlag, pp. 91-92, 1996.

Dickau, R. M. ``Stirling Numbers of the Second Kind.''

http://forum.swarthmore.edu/advanced/robertd/stirling2.html

Graham, R. L.; Knuth, D. E.; and Patashnik, O. *Concrete Mathematics: A Foundation for Computer Science, 2nd ed.*
Reading, MA: Addison-Wesley, 1994.

Knuth, D. E. ``Two Notes on Notation.'' *Amer. Math. Monthly* **99**, 403-422, 1992.

Riordan, J. *An Introduction to Combinatorial Analysis.* New York: Wiley, 1980.

Riordan, J. *Combinatorial Identities.* New York: Wiley, 1968.

Riskin, A. ``Problem 10231.'' *Amer. Math. Monthly* **102**, 175-176, 1995.

Sloane, N. J. A. Sequence A008277 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Stanley, R. P. *Enumerative Combinatorics, Vol. 1.* Cambridge, England: Cambridge University Press, 1997.

© 1996-9

1999-05-26