---

Claim. The main theorem of Yamamoto's paper states that the number $L_{k,n}$ of $k\times n$ Latin rectangles with labeled rows, columns and symbols satisfies

$$L_{k,n}\sim e^{-\binom k2}(n!)^k$$

as $n\to\infty$ whenever $k<n^{1/3-\delta}$, for a constant $\delta>0$ or more generally a positive function of $n$ tending to zero with $n^{-\delta}\to0$. This confirms the conjecture of Erdős and Kaplansky, whose theorem covers $k<(\log n)^{3/2-\epsilon}$. The proof continues their argument: the number of rows extending a given rectangle is written, in Jordan's factorial notation, as an alternating sum over quantities built from the counting function of pairs of repeated symbols, and those quantities are evaluated by classifying choices according to restricted bipartite partitions. The site records the extension under [Ya51] as the range $k\le n^{1/3-o(1)}$.

Covers. The asymptotic count for every $k<n^{1/3-\delta}$. It says nothing about larger $k$, so Problem 725, which asks for an asymptotic formula without restricting $k$, is not settled by it.

Acceptance. Refereed: Jpn. J. Math. 21 (1951), 113–119; the record gives the year only, and the page is dated to its first day. The site's curator records the theorem under [Ya51] while labeling the problem OPEN, which credits the partial result without settling the problem, so the page lists no reviewed evidence. The library card records the paper's theorem; the proof has not been reconstructed or independently reviewed in this corpus.