## Abstract

A Latin Square of order n is an n x 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 $$b^n \leq T(n) \leq c^n\sqrt{n}\,n!$$ for n â‰¥ 5, where b â‰ˆ 1.719 and c â‰ˆ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n n toroidal 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.

Translated title of the contribution | The number of transversals in a Latin square |
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Original language | English |

Pages (from-to) | 269 - 284 |

Number of pages | 16 |

Journal | Designs, Codes and Cryptography |

Volume | 40 (3) |

DOIs | |

Publication status | Published - Sep 2006 |