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The number of isomorphism classes of Latin quandles (a.k.a. left distributive quasigroups) of order n.
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%I #36 Dec 26 2021 21:09:00

%S 1,0,1,1,3,0,5,2,8,0,9,1,11,0,5,9,15,0,17,3,7,0,21,2,34,0,62,7,27,0,

%T 29,8,11,0,15,9,35,0,13,6,39,0,41,9,36,0,45

%N The number of isomorphism classes of Latin quandles (a.k.a. left distributive quasigroups) of order n.

%C A quandle is Latin if its multiplication table is a Latin square. A Latin quandle may be described as a left (or right) distributive quasigroup. Sherman Stein (see reference below) proved that a left distributive quasigroup of order n exists if and only if n is not of the form 4k + 2.

%H W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307, 2013

%H G. Ehrman, A. Gurpinar, M. Thibault, D. Yetter, <a href="http://www.math.ksu.edu/main/events/KSU-REU/REUquandle.pdf">Some Sharp Ideas on Quandle Construction</a>

%H A. Hulpke, D. Stanovský, P. Vojtěchovský, <a href="http://arxiv.org/abs/1409.2249">Connected quandles and transitive groups</a>, arXiv:1409.2249 [math.GR], 2014.

%H S. Nelson, <a href="http://arxiv.org/abs/math/0702038">A polynomial invariant of finite quandles</a>, arXiv:math/0702038 [math.QA], 2007.

%H S. K. Stein, <a href="http://dx.doi.org/10.1090/S0002-9947-1957-0094404-6">On the Foundations of Quasigroups</a>, Transactions of American Mathematical Society, 85 (1957), 228-256.

%H Leandro Vendramin, <a href="http://arxiv.org/abs/1105.5341">On the classification of quandles of low order</a>, arXiv:1105.5341v1 [math.GT].

%H Leandro Vendramin and Matías Graña, <a href="http://code.google.com/p/rig/">Rig, a GAP package for racks and quandles</a>.

%e a(2) = 0 since the only quandle of order 2 has multiplication table with rows [1,1] and [2,2].

%o (GAP) (using the Rig package)

%o LoadPackage("rig");

%o a:=[1,0];;

%o Print(1,",");

%o Print(0,",");

%o for n in [3..35] do

%o a[n]:=0;

%o for i in [1..NrSmallQuandles(n)] do

%o if IsLatin(SmallQuandle(n,i)) then

%o a[n]:=a[n]+1;

%o fi;

%o od;

%o Print(a[n],", ");

%o od; # _W. Edwin Clark_, Nov 26 2011

%Y Cf. A181769, A176077, A181771.

%Y See also Index to OEIS under quandles.

%K nonn,more

%O 1,5

%A _W. Edwin Clark_, Dec 14 2010

%E Added fact due to S. K. Stein that a(4k+2) = 0 and a reference to Stein's paper.

%E a(11)-a(35) from _W. Edwin Clark_, Nov 26 2011

%E Links to the rig Gap package by _W. Edwin Clark_, Nov 26 2011

%E a(36)-a(47) by _David Stanovsky_, Oct 01 2014