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≥n-O(n)\documentclass[12pt]{minimal}
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\begin{document}$${m\geq n-O(\sqrt{n})}$$\end{document}. An application of our Theorem 3.2 yields, for example,
s(1012,1012-2×106)/1035664464∈[1.87669,1.876982],S(1012,1012-2×106)/1035664463∈[1.30121,1.306975].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{ll}{s({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664464}} \in [1.87669, 1.876982],}\\{S({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664463}} \in [1.30121, 1.306975]}.\end{array}$$\end{document}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-tna,0≤a≤12\documentclass[12pt]{minimal}
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\begin{document}$${m = n-t {n^a}, 0 \leq a \leq \frac{1}{2}}$$\end{document}. These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range 0<a<12\documentclass[12pt]{minimal}
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\begin{document}$${0 < a < \frac{1}{2}}$$\end{document}, and they connect with a recent asymptotic expansion by Louchard for 12<a<1\documentclass[12pt]{minimal}
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\begin{document}$${\frac{1}{2} < a < 1}$$\end{document}, hence filling the gap at a=12\documentclass[12pt]{minimal}
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\begin{document}$${a = \frac{1}{2}}$$\end{document}. We also provide a generalization applicable to rook and file numbers.
