| 释义 |
Stirling Number of the First KindThe definition of the (signed) Stirling number of the first kind is a number  such that the number of permutationsof  elements which contain exactly  Cycles is
 | (1) |
 for  and  . The Generating Function is
 | (2) |
 (Wolfram Research, Champaign,IL) command StirlingS1[n,m]. The triangle of signed Stirling numbers of the first kind is(Sloane's A008275).
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 theAbsolute 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) |
which is a generalization of an Asymptotic Series for a ratio of Gamma Functions  (Gosper).See also Cycle (Permutation), Harmonic Number, Permutation, Stirling Number of the Second Kind
ReferencesAbramowitz, 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 A008275in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
|
