

A179010


The number of isomorphism classes of commutative quandles of order n.


2



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, 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, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 7
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OFFSET

1,9


COMMENTS

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 nonassociative 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.


LINKS

Table of n, a(n) for n=1..81.
V. D. Belousov, The structure of distributive quasigroups, (Russian) Mat. Sb. (N.S.) 50 (92) 1960 267298. George Glauberman, On
George Glauberman, On Loops of Odd Order II, Journal of Algebra 8 (1968), 393414.
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 3765
Gábor P. Nagy, Petr Vojtchovský, The Moufang loops of order 64 and 81, Journal of Symbolic Computation, Volume 42 Issue 9, September, 2007.
Wikipedia, Racks and quandles


CROSSREFS

Cf. A181769, A176077, A181771, A000688.
Sequence in context: A307837 A123671 A191261 * A292262 A160804 A085854
Adjacent sequences: A179007 A179008 A179009 * A179011 A179012 A179013


KEYWORD

nonn,hard,more


AUTHOR

W. Edwin Clark, Jan 04 2011


EXTENSIONS

Results due to Belousov, Nagy and Vojtchovský, and Glauberman added, and sequence extended to n = 81, by W. Edwin Clark, Jan 25 2011
In Comments section, "Every commutative quandle" replaced with "Every finite commutative quandle" by W. Edwin Clark, Mar 09 2014


STATUS

approved



