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 tex2html_wrap_inline11573 such that the number of permutations of n elements which contain exactly m cycles is

displaymath11579

This means that tex2html_wrap_inline18279 for m>n and tex2html_wrap_inline18283 . The generating function is

displaymath11581
 
 

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 tex2html_wrap_inline11583 or tex2html_wrap_inline12172 . Diagrams illustrating tex2html_wrap_inline13262tex2html_wrap_inline13264tex2html_wrap_inline13266tex2html_wrap_inline13268 , and tex2html_wrap_inline13270 (Dickau) are shown below.

tex2html_wrap13278
tex2html_wrap13280
tex2html_wrap13282
tex2html_wrap13284
tex2html_wrap13286
The Stirling numbers of the first kind satisfy the curious identity

displaymath13337

(Gosper 1996) and have the generating function

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and satisfies

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

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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.