On the Number of $6\times 7$ Double Youden Rectangles

Abstract

Results concerning the enumeration and classification of $7\times 7$ Latin squares are used to enumerate and classify all non-isomorphic Youden squares of order $6\times 7$. We show that the number of non-isomorphic Youden squares obtainable from a species of Latin square Latin Square $\delta$, depends on the number of distinct adjugate sets and the order of the automorphism group of Latin Square$\delta$. Further, we use the results obtained for $6\times 7$ Youden squares as a basis for the enumeration and classification of $6\times 7$ DYRs.