Packing of permutations into Latin squares

TL;DR

This paper explores the packing of permutations into Latin squares, establishing existence results for Latin squares with specific permutation properties and group structures for various orders, including prime and composite cases.

Contribution

It proves the existence of Latin squares that contain all permutations in rows, columns, and reverses for large n, and constructs permutation groups with specific orders for different n.

Findings

For n > 4, a set of Latin squares contains every permutation exactly once in rows, columns, and reverses.

For certain composite n, a Latin square with a permutation group of order 4n exists.

For prime n ≡ 1 mod 4, a set of mutually orthogonal Latin squares with a permutation group of order n(n-1) can be constructed.

Abstract

For every positive integer $n$ greater than $4$ there is a set of Latin squares of order $n$ such that every permutation of the numbers $1,\ldots,n$ appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If $n$ is greater than $4$ and not of the form $p$ or $2p$ for some prime number $p$ congruent to $3$ modulo $4$, then there always exists a Latin square of order $n$ in which the rows, columns, reverse rows and reverse columns are all distinct permutations of $1,\ldots,n$, and which constitute a permutation group of order $4n$. If $n$ is prime congruent to $1$ modulo $4$, then a set of $(n-1)/4$ mutually orthogonal Latin squares of order $n$ can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order $n(n-1)$.