cp's OEIS Frontend

A177886 The number of isomorphism classes of Latin quandles (a.k.a. left distributive quasigroups) of order n.

%I A177886 #36 Dec 26 2021 21:09:00
%S A177886 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 A177886 29,8,11,0,15,9,35,0,13,6,39,0,41,9,36,0,45
%N A177886 The number of isomorphism classes of Latin quandles (a.k.a. left distributive quasigroups) of order n.
%C A177886 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 A177886 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 A177886 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 A177886 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 A177886 S. Nelson, <a href="http://arxiv.org/abs/math/0702038">A polynomial invariant of finite quandles</a>, arXiv:math/0702038 [math.QA], 2007.
%H A177886 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 A177886 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 A177886 Leandro Vendramin and Matías Graña, <a href="http://code.google.com/p/rig/">Rig, a GAP package for racks and quandles</a>.
%e A177886 a(2) = 0 since the only quandle of order 2 has multiplication table with rows [1,1] and [2,2].
%o A177886 (GAP) (using the Rig package)
%o A177886 LoadPackage("rig");
%o A177886 a:=[1,0];;
%o A177886 Print(1,",");
%o A177886 Print(0,",");
%o A177886 for n in [3..35] do
%o A177886   a[n]:=0;
%o A177886   for i in [1..NrSmallQuandles(n)] do
%o A177886     if IsLatin(SmallQuandle(n,i)) then
%o A177886       a[n]:=a[n]+1;
%o A177886     fi;
%o A177886   od;
%o A177886   Print(a[n],", ");
%o A177886 od; # _W. Edwin Clark_, Nov 26 2011
%Y A177886 Cf. A181769, A176077, A181771.
%Y A177886 See also Index to OEIS under quandles.
%K A177886 nonn,more
%O A177886 1,5
%A A177886 _W. Edwin Clark_, Dec 14 2010
%E A177886 Added fact due to S. K. Stein that a(4k+2) = 0 and a reference to Stein's paper.
%E A177886 a(11)-a(35) from _W. Edwin Clark_, Nov 26 2011
%E A177886 Links to the rig Gap package by _W. Edwin Clark_, Nov 26 2011
%E A177886 a(36)-a(47) by _David Stanovsky_, Oct 01 2014