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Updated Oct 7

Problem 725

claims/: The 4 claim pages of Problem 725, one per claimant's result; the problem's standing derives from them.

Statement. Give an asymptotic formula for the number of $k\times n$ Latin rectangles.

Status. Open: the site labels the problem OPEN. Its commentary records the asymptotic $L_{k,n}\sim e^{-\binom k2}(n!)^k$ of Erdős and Kaplansky [ErKa46] for $k=o((\log n)^{3/2-\epsilon})$ and Yamamoto's extension of it to $k\le n^{1/3-o(1)}$ [Ya51], and calls sequence A001009 of the OEIS the count of such Latin rectangles. That sequence lists the normalized counts $R_{k,n}$ , rectangles whose first row and first column are in natural order, with $L_{k,n}=\dfrac{n!\,(n-1)!}{(n-k)!}\,R_{k,n}$ for the labeled count $L_{k,n}$ that the asymptotics use.

Source. erdosproblems.com/725, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #725, https://www.erdosproblems.com/725.

References.

[ErKa46] Erdős, Paul and Kaplansky, Irving, The asymptotic number of Latin rectangles. Amer. J. Math. (1946), 230-236.
[Ya51] Yamamoto, Koichi, On the asymptotic number of Latin rectangles. Jpn. J. Math. (1951), 113-119.
[GoMc90] Godsil, C. D. and McKay, B. D., Asymptotic enumeration of Latin rectangles. J. Combin. Theory Ser. B 48 (1990), no. 1, 19–44. Not in the site's bibliography; the result is named in a comment on the site's discussion thread of 2026-04-24. Not held.

Formalization. No statement file in formal-conjectures; the Li claim page links the author's Lean formalization of the sublinear-range result, which this corpus has not built.

Current assessment

The question asks for an asymptotic formula for the number $L_{k,n}$ of $k\times n$ Latin rectangles without restricting $k$ . The standing is open: no claim settles or claims to settle the full question. Three accepted partial claims, on refereed evidence, determine the asymptotic count on growing ranges of $k$ : Erdős and Kaplansky's formula $L_{k,n}\sim e^{-\binom k2}(n!)^k$ for $k<(\log n)^{3/2-\epsilon}$ [ErKa46], Yamamoto's extension of it to $k<n^{1/3-\delta}$ [Ya51], and Godsil and McKay's formula $L_{k,n}\sim(n!)^k\bigl((n)_k/n^k\bigr)^n(1-k/n)^{-n/2}e^{-k/2}$ for $k=o(n^{6/7})$ [GoMc90], the best published range. Between the 1951 range and Godsil and McKay's, Yamamoto for $k=O(n^{5/12-\epsilon})$ (Res. Rep. Sci. Div. Tokyo Womens' Univ. 19 (1969), 86–97, as Godsil and McKay cite it) and Stein for $k=o(n^{1/2})$ (J. Combin. Theory Ser. A 25 (1978), 38–49) proved $L_{k,n}\sim(n!)^k\exp\bigl(-\binom k2-k^3/(6n)\bigr)$ , the extension Erdős's 1981 survey records; both ranges lie inside Godsil and McKay's and the site credits neither, so they have no claim page. One partial claim is pending: Li's manuscript (arXiv 3 August 2026, forum 4 August 2026, written with GPT-5.6 Sol Pro and Codex) states the Godsil–McKay asymptotic for every $k=o(n)$ , with a Lean development the author reports as complete. It is not refereed, no outside reviewer has endorsed it, and this corpus has not built the Lean, so the claim is pending; it does not address $k$ of order $n$ , including the number of Latin squares, so the problem is open. No formalization beyond the author's Lean is recorded. Nothing was reconstructed in this corpus.

Search scope, 2026-10-07: the site's problem page, discussion thread and proof-claims page, the community database entry, the formal-conjectures problem listing, the OEIS entry A001009, the arXiv record of arXiv:2608.01671 and the repository README it links, the Crossref records of [ErKa46], [Ya51] and [GoMc90], and the two library cards.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (13)

Erdős and Kaplansky's asymptotic for k below a power of log n Yamamoto's asymptotic for k below n to the 1/3 Godsil and McKay's asymptotic for k = o(n^(6/7))Godsil–McKay asymptotic for all sublinear k problems/set_systems/E0725/claims Set Systems, Designs and Hypergraphs set_systems/erdos_1946_asymptotic_number_latin_rectangles Section 4 (pp. 234-236): further terms of the asymptotic series for N and f(n,k), sketched Theorem 1 (p. 232): for k < (log n)^{3/2-ε} a k-row Latin rectangle has n! e^{-k}(1 + O(n^{-c})) extensions by one row Theorem 2 (p. 234): the number of k-row Latin rectangles on n symbols is asymptotic to (n!)^k exp(-binom(k,2)) for k < (log n)^{3/2-ε}set_systems/yamamoto_1951_asymptotic_number_latin_rectangles Theorem 1 (p. 118): a Latin rectangle with k < n^{1/2-epsilon} rows has n! e^{-k}(1 + O(n^{-2 epsilon})) extensions by one row Theorem 2 (p. 118): f(n,k) ~ (n!)^k exp(-k(k-1)/2) for k < n^{1/3-delta}

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