Asymptotic enumeration of generalized Latin rectangles (Q1122579)

A \(k\times n\) Latin rectangle is a \(k\times n\) array in which each of the numbers 1,2,...,n appears exactly once in each row and at most once in each column. Let p(k,n) denote the probability that a \(k\times n\) array selected at random from the set of arrays where each row is a permutation of 1,2,...,n, is a Latin rectangle. \textit{P. Erdős} and \textit{I. Kaplansky} [Am. J. Math. 68, 230-236 (1946; Zbl 0060.028)] showed that \(p(k,n)\sim e^{-k(k-1)/2}\) as \(n\to \infty\) uniformly for \(k=o((\log n)^{3/2})\) and \textit{K. Yamamoto} [Jap. J. Math. 21, 113-119 (1951; Zbl 0045.152)] showed that the same result holds uniformly for \(k=o(n^{1/3}).\) Let B be a fixed subset of the positive integers. A \(k\times n\) array is said to be a B-Latin rectangle if each of the numbers 1,2,...,n appears exactly once in each row and each number in \(B_ n=B\cap \{1,2,...,n\}\) appears at most once in each column. Let p(B;k,n) denote the probability that a \(k\times n\) array randomly selected from the set of arrays in which each row is a permutation of \(\{\) 1,2,...,n\(\}\), is a B-Latin rectangle. Using a generalization of methods of \textit{C. M. Stein} [J. Comb. Theory, Ser. A 25, 38-49 (1978; Zbl 0429.05022)], the author generalizes Yamamoto's result by proving: If \((| B_ n|)^{- 1}=O(n^{-\alpha})\) for \(1/2<\alpha \leq 1\), then \((p(B;k,n))^{n/| B_ n|}\sim e^{-k(k-1)/2}\) as \(n\to \infty\), uniformly for \(k=o(n^{(2\alpha -1)/3}).\)