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%I #35 Jun 15 2022 06:22:27
%S 1,0,1,0,1,0,1,0,2,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,2,0,3,0,1,0,1,0,1,0,
%T 1,0,1,0,1,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,1,0,1,0,1,0,1,0,2,0,1,0,1,0,
%U 1,0,1,0,1,0,2,0,1,0,1,0,7
%N The number of isomorphism classes of commutative quandles of order n.
%C A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.
%H V. D. Belousov, <a href="http://mi.mathnet.ru/eng/msb/v92/i3/p267">The structure of distributive quasigroups</a>, (Russian) Mat. Sb. (N.S.) 50 (92) 1960 267-298.
%H George Glauberman, <a href="http://dx.doi.org/10.1016/0021-8693(68)90050-1">On Loops of Odd Order II</a>, Journal of Algebra 8 (1968), 393-414.
%H David Joyce, <a href="http://dx.doi.org/10.1016/0022-4049(82)90077-9">A classifying invariant of knots, the knot quandle</a>, J. Pure Appl. Algebra 23 (1982) 37-65
%H Gábor P. Nagy and Petr Vojtchovský, <a href="http://dx.doi.org/10.1016/j.jsc.2007.06.004">The Moufang loops of order 64 and 81</a>, Journal of Symbolic Computation, Volume 42 Issue 9, September, 2007.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Quandles">Racks and quandles</a>
%Y Cf. A181769, A176077, A181771, A000688.
%K nonn,hard,more
%O 1,9
%A _W. Edwin Clark_, Jan 04 2011
%E Results due to Belousov, Nagy and Vojtchovský, and Glauberman added, and sequence extended to n = 81, by _W. Edwin Clark_, Jan 25 2011
%E In Comments section, "Every commutative quandle" replaced with "Every finite commutative quandle" by _W. Edwin Clark_, Mar 09 2014
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