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 2 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 non-trisotopic 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.