Packing of permutations into Latin squares

Stephan Foldes,András Kaszanyitzky,László Major

doi:10.1016/j.dam.2021.03.001

Stephan Foldes, András Kaszanyitzky + Show 1 more

Open Access

https://doi.org/10.1016/j.dam.2021.03.001

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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,…,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,…,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).

Highlights

A permutation is a bijective map f from a finite, n-element set to itself
When the domain is fixed, or it is an arbitrary n-element set, the group of all permutations on that domain is denoted by Sn
N-element set R, a set S of m Latin squares of order n with entries in R is called a packing if (i) all members of S are strongly asymmetric, (ii) no line of any member of S appears as a line in any other member of S

Summary

Introduction

A permutation is a bijective map f from a finite, n-element set (the domain of the permutation) to itself. Without loss of generality we shall throughout this paper assume that matrix entries and the elements of the underlying sets of permutations belong to some nontrivial ring with a zero and a unit element 1 This will allow in particular the multiplication of any square matrix of order n with the reversal matrix ( called the exchange matrix ) of order n given by J(i, j) = δi,n+1−j (where δi,j is the Kronecker delta). N-element set R, a set S of m Latin squares of order n with entries in R (meaning that each line of each member of S represents a permutation of R) is called a packing if (i) all members of S are strongly asymmetric, (ii) no line of any member of S appears as a line in any other member of S.

Packing into Latin squares of odd order

Packing into Latin squares of even order

Packing a permutation group into a single Latin square

Mutually orthogonal Latin squares

Minimum number of lines and symmetries of Latin squares

A BJ JB JAJ