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Statement

Setting (p. 113). $f(n,k)$ is the number of $n$ by $k$ Latin rectangles in the symbols $1,2,\ldots,n$, that is, of $k$ rows of length $n$ with no symbol repeated in a row or in a column; the rows, columns and symbols are labeled.

Theorem 2 (p. 118, quoted). "For $k<n^{1/3-\delta}$, $\delta$ being positive constant, we have the asymptotic relation

$$f(n,k)\sim(n!)^k\exp(-k(k-1)/2)$$

for the number $f(n,k)$ of $n$ by $k$ Latin rectangles."

The relation is as $n\to\infty$, uniformly in the $k$ allowed: the proof gives $(1-cn^{-2\varepsilon})^k<\exp(k(k-1)/2)\,(n!)^{-k}f(n,k)<(1+cn^{-2\varepsilon})^k$ with $\varepsilon=1/6+\delta$ for all sufficiently large $n$, and both bounds tend to $1$.

Remark (p. 119). The paper says the generalization made for Theorem 1 is immediate here: $\delta$ may be a positive function of $n$ with $n^{-\delta}\to0$ as $n\to\infty$. The introduction (p. 113) states the result in this form, for $\delta$ a positive constant "or more generally, may be a positive-valued function of $n$ tending to zero such that" $n^{-\delta}\to0$.

Erdős and Kaplansky had proved the same relation for $k<(\log n)^{3/2-\varepsilon}$ and conjectured it for $k$ up to nearly $n^{1/3}$ (p. 113); Theorem 2 confirms that conjecture. The paper adds (p. 119) that the range $k<n^{1/3-\delta}$ is the limit of its method, as seen from (27), and that it seems likely to be the natural boundary of the problem, as Erdős and Kaplansky observed.

Proof pointer

Pp. 118--119. Apply Theorem 1 with $\varepsilon=1/6+\delta$ to the extensions of $n$ by $i$ Latin rectangles for $i=0,1,\ldots,k-1$, where $i<n^{1/3-\delta}=n^{1/2-\varepsilon}$: every $n$ by $i$ Latin rectangle has between $n!\,e^{-i}(1-cn^{-2\varepsilon})$ and $n!\,e^{-i}(1+cn^{-2\varepsilon})$ extensions, so $1-cn^{-2\varepsilon}<e^i(n!)^{-1}f(n,i+1)/f(n,i)<1+cn^{-2\varepsilon}$, with $f(n,0)=1$. Multiplying these $k$ inequalities gives the bounds above, and $k\log(1+cn^{-2\varepsilon})<cn^{-2(1/6+\delta)}n^{1/3-\delta}=cn^{-3\delta}$ (27) shows they tend to $1$.

Read depth

Claims checked: the setting, Theorem 2, the remark after it and the introduction's statement were read clause by clause on the page images of the print, and the proof on pp. 118--119 was followed. Nothing here is independently reviewed.

Dependencies

• Theorem 1 (p. 118), the per-row extension estimate.

Source. K. Yamamoto, On the asymptotic number of Latin rectangles, Jpn. J. Math. 21 (1951), 113--119, doi:10.4099/jjm1924.21.0_113; the edition read is named on the source card.

Bears on

• Problem 725: Theorem 2 gives the asymptotic number $f(n,k)\sim(n!)^k\exp(-k(k-1)/2)$ of $n$ by $k$ Latin rectangles, the problem's $k\times n$ Latin rectangles, for $k<n^{1/3-\delta}$, and by the remark for $\delta$ a positive function of $n$ with $n^{-\delta}\to0$. It says nothing about larger $k$, which the problem also asks about.