## Stirling Triangle for Cycles

StirlingTriangle for Cycles

11
24  50  35  10
120  274  225  85  15
720  1764  1624  735  175  21
5040  13068  13132  6769  1960  322  28
40320  109584  118124  67284  22449  4536  546  36

Reference: Graham, Knuth and Patashnik, Concrete Mathematics 6.1

The definition of the (signed) Stirling number of the first kind is a number  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.

The Stirling numbers of the first kind satisfy the curious identity

(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.

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