Erdős Problem #725
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`M` is a \(k\times n\) Latin rectangle: `M i j` is the entry in row `i` and column `j`, the entries are in `Fin n` (standing for \(\{1,\dots,n\}\)), every row is a permutation of the entries, and no entry is repeated in a column.
\(L(k,n)\) is the number of \(k\times n\) Latin rectangles. Rows, columns and entries are labelled and the rows are ordered. It is \(0\) for \(k>n\geq 1\) and \(1\) for \(k=0\).
Attempts
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verified Godsil--McKay through \(k=o(n^{6/7})\), proved a uniform permanent reduction/logarithmic bound, and independently computed exact reduced counts through \(n=6\) plus \(R(4,7)=1{,}293{,}216\);
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