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Statement

Setting (p. 113). An $n$ by $k$ Latin rectangle $L$ is a Latin rectangle in the symbols $1,2,\ldots,n$ with $k$ rows of length $n$ (no symbol repeated in a row or in a column). $N$ is the number of ways to adjoin a $(k+1)$th row to $L$ so that the result is an $n$ by $(k+1)$ Latin rectangle.

Theorem 1 (p. 118, quoted). "For $k<n^{1/2-\varepsilon}$, $\varepsilon$ being positive constant, the inequality

$$|Ne^k/n!-1|<cn^{-2\varepsilon},\qquad c\text{: absolute constant,}$$

is valid for sufficiently large $n$."

The bound is uniform over the rectangle: $N$ depends on $L$, but the proof bounds the error using only $n$ and $k$, so it holds for every $n$ by $k$ Latin rectangle $L$.

Remark (p. 118). The paper notes that $\varepsilon$ need not be constant: it may be a positive function of $n$ with $n^{-\varepsilon}\to0$ as $n\to\infty$, with the proof unchanged; for instance $\varepsilon=\log^{(m)}n/\log n$, where $\log^{(m)}$ is the $m$ times iterated logarithm and $m$ is a fixed positive integer.

The paper adds (p. 119) that the range $k<n^{1/2-\varepsilon}$ is the limit of its method, as seen from its estimate (25).

Proof pointer

Pp. 113--118. Starting from the Erdős--Kaplansky inclusion--exclusion formula (5) for $N$, the paper rewrites it as (6) (p. 114), $N=n!\sum_{t=0}^{n}(-1)^tG_t\sigma_{n-t}/(n)_t$, where $\sigma_m=\sum_{u=0}^{m}(-k)^u/u!$, $(n)_t=n!/(n-t)!$, and $G_t=\sum_s(-1)^sF(s,t)$ with $F(s,t)$ the number of ways to choose $s$ pairs of equal symbols using all of $t$ entries in different columns of $L$; $G_0=1$ and $G_1=0$. Grouping the choices by the multiplicities of the symbols involved (a bipartite partition $t=\sum_i ia_i$, $u=\sum_i a_i$), Lemma 1 (p. 115) evaluates the sign-weighted count for a single symbol occurring $m$ times as $(-1)^{m-1}(m-1)$, and a crude count of the entry choices (18) bounds $G_t$ by (19) with weights $B(\pi)$, whose total over all unrestricted such partitions is $e$ by Lemma 2 (p. 117). Approximating $\sigma_m$ by $e^{-k}$ gives (23), and Stirling's formula bounds the two resulting sums by $2e^2n^{-2\varepsilon}$ each when $k<n^{1/2-\varepsilon}$ (p. 118).

Read depth

Claims checked: the setting, Theorem 1 and the remark after it were read clause by clause on the page images of the print, and the proof on pp. 113--118 was followed at the level of the pointer above. Nothing here is independently reviewed.

Dependencies

None in the corpus. The external input is the Erdős--Kaplansky formula (5) for $N$ (Amer. J. Math. 68 (1946), 230--236), which the paper takes as its starting point.

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 1 is the per-row estimate from which the paper derives Theorem 2, the asymptotic count of Latin rectangles for $k<n^{1/3-\delta}$; on its own it counts one-row extensions and gives no count of rectangles.