All-even Latin squares
Each row of a Latin square of order \(n\) or of a \(k\times n\) Latin rectangle with \(k\leq n\) can be thought of as a permutation of the elements in \(\{1,2,\dots,n\}\). The authors show that the proportion of Latin squares of order \(n\) for which all rows are even permutations is at most \(c^n\), where \(\sqrt{{3\over 4}}<c<1\). Coupled with work by Cameron and Kantor, this result establishes that for almost all Latin squares of order \(n\), the permutations given by the rows generate the symmetric group \(S_n\). The authors also show that for \(k\times n\) Latin rectangles with \(k\leq n-7\), the proportion of Latin rectangles for which all rows are even permutations is asymptotically equal to \(2^{-k}\), thereby indicating a ``quasi-random'' behavior for the rows.
scientific article; zbMATH DE number 935380