부산대학교 도서관

A Latin Square of order nis an n×  narray of nsymbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of nentries, one selected from each row and each column of a Latin Square of order nsuch 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 $$b^n \leq T(n) \leq c^n\sqrt{n}\,n!$$for n≥ 5, where b≈ 1.719 and c≈ 0.614. A corollary of this result is an upper bound on the number of placements of nnon-attacking queens on an n×  ntoroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.A Latin Square of order nis an n×  narray of nsymbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of nentries, one selected from each row and each column of a Latin Square of order nsuch 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 $$b^n \leq T(n) \leq c^n\sqrt{n}\,n!$$for n≥ 5, where b≈ 1.719 and c≈ 0.614. A corollary of this result is an upper bound on the number of placements of nnon-attacking queens on an n×  ntoroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.