Latin Squares: Transversals and counting of Latin squares
نویسندگان
چکیده
Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a set of n distinct entries such that no two entries share the same row, column or symbol. This is an example of mutually orthogonal latin squares. L1= 2 3 1 1 2 3 3 1 2
منابع مشابه
More mutually orthogonal Latin squares
A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. In this paper we give some constructions of pairwise orthogonal diagonal Latin squares. As an application of such constructions we obtain some new infinite classes of pairwise orthogonal diagonal Latin squares which are useful in the study of pairwise orthogonal diagonal Latin squares.
متن کاملTransversals in Latin Squares: A Survey
A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries containing no pair of entries that share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutuall...
متن کاملOn the existence of self-orthogonal diagonal Latin squares
A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. A Latin square is self-orthogonal if it is orthogonal to its transpose. In an earlier paper Danhof, Phillips and Wallis considered the question of the existence of self-orthogonal diagonal Latin squares of order 10. In this paper we shall present some constructions of self-orthogonal diagonal ...
متن کاملThe number of transversals in a Latin square
A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that bn ≤ T (n) ≤ ...
متن کاملTransversals in Latin Squares
A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogon...
متن کامل