### Abstract

Let L be a Latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials det(L) and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 Latin subsquares in L can be determined from both det(L) and per(L). More generally, we can diagnose from det(L) or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in det(L) and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic Latin squares with equal permanents and equal determinants exist for all orders n≥9 that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.

Original language | English |
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Pages (from-to) | 132-148 |

Number of pages | 17 |

Journal | Journal of Combinatorial Designs |

Volume | 24 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Mar 2016 |

### Keywords

- determinant
- Latin square
- permanent
- quasigroup
- transversal

## Cite this

*Journal of Combinatorial Designs*,

*24*(3), 132-148. https://doi.org/10.1002/jcd.21418