A note on the asymptotic number of Latin rectangles (Q1266379)

The author uses two inequalities from the theory of permanents, ``Minc's conjecture'' and ``van der Waerden's conjecture'' to study \(L(k, n)\), the number of \(k\times n\) Latin rectangles. He gives a new proof for the asymptotic formula of \textit{C. Stein} [J. Comb. Theory, Ser. A 25, 38-49 (1978; Zbl 0429.05022)] \[ L(k,n)\sim (n!)^k e^{-{k\choose 2}-{k^3\over 6n}} \] on a slightly narrower range then the original---\(k= o(\sqrt{n/\log n})\) instead of \(k= o(\sqrt n)\). In addition, although no asymptotic formula is obtained for \(L(n, n)\), the following new estimate is given: \[ \{L(n, n)\}^{n^{-1-\varepsilon}}\sim e^{-2n^{1- \varepsilon}} n^{n^{1- \varepsilon}}. \]