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Statement

Setting (p. 230, Section 2). An $n$ by $k$ Latin rectangle has $k$ rows, each an arrangement of $1,\ldots,n$, with distinct integers in each column (the printed definition says $n$ rows and $k$ columns, but puts $1,\ldots,n$ in each row and adds rows one at a time; see Theorem 1). The paper writes ${}_kC_2$ for $\binom k2$.

Theorem 2 (p. 234). Let $f(n,k)$ be the number of $n$ by $k$ Latin rectangles and suppose $k<(\log n)^{3/2-\epsilon}$. Then

$$f(n,k)\,(n!)^{-k}\exp\binom k2\to1 \quad\text{as } n\to\infty. \qquad (18)$$

So $f(n,k)\sim(n!)^k e^{-\binom k2}$ in that range, with $k$ fixed or growing with $n$; $\epsilon$ is a fixed positive number, as in Theorem 1. The introduction (p. 230) calls this formula an easy heuristic conjecture, previously proved for $k=3$, and says the paper proves it for $k$ fixed and for $k<(\log n)^{3/2-\epsilon}$.

Proof pointer

P. 234. By Theorem 1, applied to each $i$-row rectangle, $f(n,i+1)$ lies between $f(n,i)\,n!\,e^{-i}(1\pm n^{-c})$. Multiplying from $i=1$ to $k-1$ puts $f(n,k)$ between $(n!)^k\exp(-\binom k2)(1\pm n^{-c})^k$, and both $(1+n^{-c})^k$ and $(1-n^{-c})^k$ tend to $1$ because $k$ is at most a power of $\log n$.

Read depth

Claims checked: Theorem 2 and its proof were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

• Theorem 1 (p. 232), the one-row estimate.

Source. P. Erdős and I. Kaplansky, The asymptotic number of Latin rectangles, Amer. J. Math. 68 (1946), no. 2, 230--236, doi:10.2307/2371834; the edition read is named on the source card.

Bears on

• Problem 725: the problem asks for an asymptotic formula for the number of $k\times n$ Latin rectangles without restricting $k$; Theorem 2 gives $f(n,k)\sim(n!)^k e^{-\binom k2}$ for $k<(\log n)^{3/2-\epsilon}$ and says nothing about larger $k$. The problem's claim page for this paper records that partial answer.