On Even and Odd Latin Squares

Latin squares can be classified as odd or even according to the signs of the permutations given by their rows and columns. In this paper, the behaviour of the parities of a latin square under the action of the isotopy group (permuting rows, columns, and symbols) and the transformation group (interchanging rows, columns, and symbols) is analyzed. A rule is given that shows that the behaviour of parities under the action of the transformation group is completely determined, and only depends on the order n. Also, we give a short proof of the equivalence of the conjectures of Alon-Tarsi and Huang-White and present the results of a computer calculation that show that the number of even and odd latin squares of order 8 are not equal, thus proving the Alon-Tarsi conjecture for this particular order.