StirlingTriangle for Cycles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | |||||||||

2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

3 | 2 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |

4 | 6 | 11 | 6 | 1 | 0 | 0 | 0 | 0 | 0 | |

5 | 24 | 50 | 35 | 10 | 1 | 0 | 0 | 0 | 0 | |

6 | 120 | 274 | 225 | 85 | 15 | 1 | 0 | 0 | 0 | |

7 | 720 | 1764 | 1624 | 735 | 175 | 21 | 1 | 0 | 0 | |

8 | 5040 | 13068 | 13132 | 6769 | 1960 | 322 | 28 | 1 | 0 | |

9 | 40320 | 109584 | 118124 | 67284 | 22449 | 4536 | 546 | 36 | 1 |

Reference: Graham, Knuth and Patashnik, Concrete Mathematics 6.1

The definition of the (signed)

This means that
for *m*>*n* and
. The **generating function** is

The nonnegative version simply gives the number of permutations of *n*
objects having *m* cycles and is obtained by taking the absolute value
of the signed version. It is denoted
or
. Diagrams illustrating
,
,
,
, and
(Dickau) are shown below.

(Gosper 1996) and have the generating function

and satisfies

The Stirling numbers can be generalized to non-integral arguments (``Stirling polynomials?'') using the identity,

which is a generalization of an asymptotic series for gamma functions
(Gosper).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Stirling
Numbers of the First Kind.'' §24.1.3 in *Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
9th printing.* New York: Dover, p. 824, 1972.

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 First Kind.'' http://forum.swarthmore.edu/advanced/robertd/stirling1.html.