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) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2036029552582883134196099"> <mml:semantics> <mml:mn>2036029552582883134196099</mml:mn> <mml:annotation encoding="application/x-tex">2036029552582883134196099</mml:annotation> </mml:semantics> </mml:math> </inline-formula> main classes of Latin squares of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="11"> <mml:semantics> <mml:mn>11</mml:mn> <mml:annotation encoding="application/x-tex">11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="semicolon"> <mml:semantics> <mml:mo>;</mml:mo> <mml:annotation encoding="application/x-tex">;</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (ii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6108088657705958932053657"> <mml:semantics> <mml:mn>6108088657705958932053657</mml:mn> <mml:annotation encoding="application/x-tex">6108088657705958932053657</mml:annotation> </mml:semantics> </mml:math> </inline-formula> isomorphism classes of one-factorizations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript 11 comma 11"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> <mml:mo>,</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">K_{11,11}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="semicolon"> <mml:semantics> <mml:mo>;</mml:mo> <mml:annotation encoding="application/x-tex">;</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (iii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="12216177315369229261482540"> <mml:semantics> <mml:mn>12216177315369229261482540</mml:mn> <mml:annotation encoding="application/x-tex">12216177315369229261482540</mml:annotation> </mml:semantics> </mml:math> </inline-formula> isotopy classes of Latin squares of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="11"> <mml:semantics> <mml:mn>11</mml:mn> <mml:annotation encoding="application/x-tex">11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="semicolon"> <mml:semantics> <mml:mo>;</mml:mo> <mml:annotation encoding="application/x-tex">;</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (iv) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1478157455158044452849321016"> <mml:semantics> <mml:mn>1478157455158044452849321016</mml:mn> <mml:annotation encoding="application/x-tex">1478157455158044452849321016</mml:annotation> </mml:semantics> </mml:math> </inline-formula> isomorphism classes of loops of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="11"> <mml:semantics> <mml:mn>11</mml:mn> <mml:annotation encoding="application/x-tex">11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="semicolon"> <mml:semantics> <mml:mo>;</mml:mo> <mml:annotation encoding="application/x-tex">;</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and (v) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="19464657391668924966791023043937578299025"> <mml:semantics> <mml:mn>19464657391668924966791023043937578299025</mml:mn> <mml:annotation encoding="application/x-tex">19464657391668924966791023043937578299025</mml:annotation> </mml:semantics> </mml:math> </inline-formula> isomorphism classes of quasigroups of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="11"> <mml:semantics> <mml:mn>11</mml:mn> <mml:annotation encoding="application/x-tex">11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The enumeration is constructive for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1151666641"> <mml:semantics> <mml:mn>1151666641</mml:mn> <mml:annotation encoding="application/x-tex">1151666641</mml:annotation> </mml:semantics> </mml:math> </inline-formula> main classes with an autoparatopy group of order at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .