The number of Latin squares of order 11

Abstract

Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) $2036029552582883134196099$ main classes of Latin squares of order $11$$;$ (ii) $6108088657705958932053657$ isomorphism classes of one-factorizations of $K_{11,11}$$;$ (iii) $12216177315369229261482540$ isotopy classes of Latin squares of order $11$$;$ (iv) $1478157455158044452849321016$ isomorphism classes of loops of order $11$$;$ and (v) $19464657391668924966791023043937578299025$ isomorphism classes of quasigroups of order $11$. The enumeration is constructive for the $1151666641$ main classes with an autoparatopy group of order at least $3$.