The number of Latin squares of order 11

TL;DR

This paper provides an exact enumeration of Latin squares of order 11, along with related combinatorial objects, using both constructive and nonconstructive methods, significantly advancing the understanding of their classification.

Contribution

It presents the first complete enumeration of Latin squares of order 11 and related structures, including isomorphism and isotopy classes, using novel combinatorial techniques.

Findings

Over 2 quintillion Latin squares of order 11

Precise counts of related combinatorial objects like loops and quasigroups

Constructive enumeration for classes with large autoparatopy groups

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.