LAUSR

Paratopism is a well-known action of the wreath product Sn≀S3 on Latin squares of order n. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let Par(n) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order n. We prove a number of general properties of autoparatopisms. Applying these results, we determine Par(n) for n⩽17. We also study the proportion of all paratopisms that are in Par(n) as n→∞.