Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completely effective error bounds for Stirling Numbers of the first and
  second kind via Poisson Approximation   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.

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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th