2014/04/11 by Richard Arratia, Arratia, Richard, Stephen DeSalvo +1
Mathematics · #05A16 #05A20 #11B73 #60C05 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A16 #msc:05A20 #msc:11B73 #msc:60C05
paper · pdf · doi:10.48550/arxiv.1404.3007
19 pages, 4 Figures, 19 References
arxiv created 2016/09/09 · arxiv updated 2016/09/12
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 ≥ n - O(√(n)). An application of our Theorem 5 yields, for example, s(1012, 1012-2× 106)/1035664464 ∈ [ 1.87669, 1.876982 ], S(1012, 1012-2× 106)/1035664463 ∈ [ 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 na, 0 ≤ a ≤ (1)/(2). These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range 0<a<(1)/(2), and they connect with a recent asymptotic expansion by Louchard for (1)/(2)<a < 1, hence filling the gap at a = (1)/(2). We also provide a generalization applicable to rook and file numbers.