Permanents and Determinants of Latin Squares

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.