Parity Types, Cycle Structures and Autotopisms of Latin Squares

The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A fast algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented.