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Annals of Combinatorics

, Volume 21, Issue 1, pp 1–24 | Cite as

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

  • Richard Arratia
  • Stephen DeSalvoEmail author
Article
  • 80 Downloads

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 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA

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