Packing of permutations into Latin squares (Q2022513)

A finite set \(S\) of \(m\) Latin squares of order \(n\) with entries in a given set of \(n\) distinct symbols is said to be a packing if (a) all the Latin squares in \(S\) are strongly asymmetric (that is, the \(4n\) sequences of length \(n\) formed by its rows, columns, reverse rows and reverse columns are pairwise distinct); and (b) none of these sequences appears twice in \(S\). In such a case, the \(4mn\) distinct permutations described by all the sequences in \(S\) are said to be packed in \(S\). In this paper, the authors prove in a constructive way that the symmetric group \(S_n\), with \(n\geq 5\) can always be packed into some set of strongly asymmetric Latin squares. It is also constructively proven that, if \(n>4\) is not of the form \(p\) or \(2p\), with prime \(p\) congruent to 3 modulo 4, then the symmetric group \(S_n\) has a subgroup of order \(4n\) that can be packed into a single strongly asymmetric Latin square of order \(n\). Moreover, no permutation group can be packed into a single asymmetric Latin square of a prime order congruent to 3 modulo 4. Finally, if \(n\) is prime congruent to \(1\) modulo 4, then the symmetric group \(S_n\) has a subgroup of order \(n(n - 1)\) that can be packed into a set of \((n - 1)/4\) mutually orthogonal Latin squares of order \(n\). The authors also deal with the existing relationship among the parity of the order of a Latin square, some symmetry properties of the latter and the minimum number of distinct sequences described by its rows, columns, reverse rows and reverse columns.