The definition of the (signed) Stirling number of the first kind is a number such that the number of permutations
of elements which contain exactly Cycles is

(1) |

(2) |

The Nonnegative version simply gives the number of Permutations of objects having Cycles (with cycles in opposite directions counted as distinct) and is obtained by taking the Absolute Value of the signed version. The nonnegative Stirling number of the first kind is denoted or . Diagrams illustrating , , , and (Dickau) are shown below.

The nonnegative Stirling numbers of the first kind satisfy the curious identity

(3) |

(4) |

(5) |

(6) |

which is a generalization of an Asymptotic Series for a ratio of 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.

Adamchik, V. ``On Stirling Numbers and Euler Sums.'' *J. Comput. Appl. Math.* **79**, 119-130, 1997.

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.

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

Sloane, N. J. A. Sequence A008275 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-26