Annals of Combinatorics

, Volume 21, Issue 1, pp 1–24

# Completely Effective Error Bounds for Stirling Numbers of the First and Second Kinds via Poisson Approximation

• Richard Arratia
• Stephen DeSalvo
Article

## Abstract

We provide completely effective error estimates for Stirling numbers of the first and second kinds, denoted by s(n, m) and S(n, m), respectively. These bounds are useful for values of $${m\geq n-O(\sqrt{n})}$$. An application of our Theorem 3.2 yields, for example,
$$\begin{array}{ll}{s({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664464}} \in [1.87669, 1.876982],}\\{S({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664463}} \in [1.30121, 1.306975]}.\end{array}$$
The bounds are obtained via Chen-Stein Poisson approximation, using an interpretation of Stirling numbers as the number of ways of placing non-attacking rooks on a chess board. As a corollary to Theorem 3.2, summarized in Proposition 2.4, we obtain two simple and explicit asymptotic formulas, one for each of s(n, m) and S(n, m), for the parametrization $${m = n-t {n^a}, 0 \leq a \leq \frac{1}{2}}$$. These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range $${0 < a < \frac{1}{2}}$$, and they connect with a recent asymptotic expansion by Louchard for $${\frac{1}{2} < a < 1}$$, hence filling the gap at $${a = \frac{1}{2}}$$. We also provide a generalization applicable to rook and file numbers.

## Keywords

Stirling numbers of the first kind Stirling numbers of the second kind Poisson approximation Stein’s method asymptotic enumeration of combinatorial sequences completely effective error estimates rook numbers file numbers

## Mathematics Subject Classification

05A16 60C05 11B73

## References

1. 1.
Chelluri R., Richmond L., Temme N.: Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20(1), 1–13 (2000)
2. 2.
Flajolet P., Prodinger H.: On Stirling numbers for complex arguments and Hankel contours. SIAM J. Discrete Math. 12(2), 155–159 (1999)
3. 3.
Garsia A.M., Remmel J.B.: Q-counting rook configurations and a formula of Frobenius. J. Combin. Theory Ser. A 41(2), 246–275 (1986)
4. 4.
Good I.J.: Saddle-point methods for the multinomial distribution. Ann. Math. Statistics 28, 861–881 (1957)
5. 5.
Good I.J.: An asymptotic formula for the differences of the powers at zero. Ann. Math. Statistics 32, 249–256 (1961)
6. 6.
Grimmett G.R., Stirzaker D.R.: Probability and Random Processes. Third edition. Oxford University Press, New York (2001)
7. 7.
Hsu L.C.: Note on an asymptotic expansion of the nth difference of zero. Ann. Math. Statistics 19, 273–277 (1948)
8. 8.
Hwang H.-K.: Asymptotic expansions for the Stirling numbers of the first kind. J. Combin. Theory Ser. A 71(2), 343–351 (1995)
9. 9.
Jordan, C.: Cours d’analyse de l’École polytechnique. Tome I. Les Grands Classiques Gauthier-Villars. [Gauthier-VillarsGreat Classics]. Éditions Jacques Gabay, Sceaux, 1991. Calcul différentiel. [Differential calculus], Reprint of the third (1909) editionGoogle Scholar
10. 10.
Kaplansky I., Riordan J.: The problem of the rooks and its applications. Duke Math. J. 13, 259–268 (1946)
11. 11.
Louchard G.: Asymptotics of the Stirling numbers of the first kind revisited: a saddle point approach. Discrete Math. Theor. Comput. Sci. 12(2), 167–184 (2010)
12. 12.
Louchard G.: Asymptotics of the Stirling numbers of the second kind revisited. Appl. Anal. Discrete Math. 7(2), 193–210 (2013)
13. 13.
Moser L., Wyman M.: Stirling numbers of the second kind. Duke Math. J. 25, 29–43 (1957)
14. 14.
Moser L., Wyman M.: Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, 133–146 (1958)
15. 15.
Remmel, J.B., Wachs, M.L.: Rook theory, generalized Stirling numbers and (p, q)-analogues. Electron. J. Combin. 11(1), #R84 (2004)Google Scholar
16. 16.
Sachkov, V.N.: Probabilistic Methods in Combinatorial Analysis. Encyclopedia of Mathematics and its Applications, Vol. 56. Translated from the Russian, Revised by the author. Cambridge University Press, Cambridge (1997)Google Scholar
17. 17.
Salvy B., Shackell J.: Symbolic asymptotics: multiseries of inverse functions. J. Symbolic Comput. 27(6), 543–563 (1999)
18. 18.
Stanley, R.P.: Enumerative Combinatorics. Volume 1. Second edition. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2012)Google Scholar
19. 19.
Wilf H.S.: The asymptotic behavior of the Stirling numbers of the first kind. J. Combin. Theory Ser. A 64(2), 344–349 (1993)