| 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 | |
such that the number of permutations of n elements which contain
exactly m cycles is
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.