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Claim. Godsil and McKay prove that the number $L_{k,n}$ of $k\times n$ Latin rectangles with labeled rows, columns and symbols satisfies

$$L_{k,n}\sim(n!)^k\Bigl(\frac{(n)_k}{n^k}\Bigr)^n \Bigl(1-\frac kn\Bigr)^{-n/2}e^{-k/2}$$

as $n\to\infty$ with $k=o(n^{6/7})$, where $(n)_k=n(n-1)\cdots(n-k+1)$. The method counts the one-row extensions of a rectangle $R$ as the perfect matchings of $K_{n,n}$ that avoid the $k$-regular bipartite graph $G(R)$ of $R$, writes that number as $\int_0^\infty e^{-x}r(G,x)\,dx$ with $r(G,x)$ the rook polynomial of $G$, whose zeros lie in $[0,4k-4]$, expands it in $k$ and the counts of small subgraphs of $G$, chiefly $4$-cycles, averages over a random $k\times n$ rectangle, and multiplies the average one-row ratios. On the range $k<n^{1/3-\delta}$ the formula agrees with the asymptotic $e^{-\binom k2}(n!)^k$ of Erdős and Kaplansky and Yamamoto, and beyond it the extra factors are no longer asymptotically $1$. The result was announced in Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 91–92, linked above. The site's commentary does not name the paper; a comment on the site's discussion thread of 2026-04-24 asks for it to be added and states the formula with its range. The paper is not held in this corpus, and the account above follows the paper's abstract and introduction, the 1984 announcement and the thread's statement of the theorem.

Covers. The asymptotic count for every $k=o(n^{6/7})$. It says nothing about larger $k$, so Problem 725, which asks for an asymptotic formula without restricting $k$, is not settled by it; Li's manuscript claims the same formula for every $k=o(n)$.

Acceptance. Refereed: C. D. Godsil and B. D. McKay, Asymptotic enumeration of Latin rectangles, J. Combin. Theory Ser. B 48 (1990), no. 1, 19–44; the page is dated by the result's first posting, the announcement in the January 1984 issue of Bull. Amer. Math. Soc. (N.S.). The site's curator does not credit the result, and the site labels the problem OPEN, so the page lists no reviewed evidence. The proof has not been reconstructed or independently reviewed in this corpus.