Completely effective error bounds for Stirling Numbers of the first and second kind via Poisson Approximation

Abstract

We provide completely effective error estimates for Stirling numbers of the first and second kind, 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 5 yields, for example, \[ s(10^{12},\ 10^{12}-2\times 10^6)/10^{35664464} \in [ 1.87669, 1.876982 ], \] \[ S(10^{12},\ 10^{12}-2\times 10^6)/10^{35664463} \in [ 1.30121, 1.306975 ]. \] 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 5, summarized in Proposition 1, 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.