E    
  D  
    T

Encyclopaedia of DesignTheory: Latin squares

General Partition Incidence Array
Experimental Other designs Math properties Stat properties
Server External Bibliography Miscellanea

Designs related to Latin squares

Among special cases of Latin squares are the Cayley tables of groups.

Combinatorial structures equivalent to Latin squares include nets of degree 3, and orthogonal arrays of strength 2 and index 1. Equivalent algebraic structures are quasigroups and loops.

A Latin square is also equivalent to a transversal design having three groups of size n and n² blocks of size 3, so that two points in different groups lie in a unique block.

Further examples include 1-factorisations of complete bipartite graphs, maximum-size sets of mutually non-attacking rooks on a cubic chessboard, and sharply transitive sets of permutations. We refer to the topic essay for further details of these structures and the notions of equivalence of Latin squares that they produce.

Related structures include MOLS and nets of arbitrary degree, as well as more general orthogonal arrays.

Other generalisations include semi-Latin squares and SOMAs.

The rows of a Latin square of order n form a sharply transitive set of permutations of {1,...,n}: this means that, for any two symbols i and j, there is a unique permutation in the set which carries i to j. A more general concept is that of a sharply t-transitive set of permutations, containing a unique element mapping any t-tuple of distinct points to any other.


Table of contents | Glossary | Topics | Bibliography | History

Peter J. Cameron
5 July 2006