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 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.
LINKS
V. D. Belousov, The structure of distributive quasigroups, (Russian) Mat. Sb. (N.S.) 50 (92) 1960 267-298.
George Glauberman, On Loops of Odd Order II, Journal of Algebra 8 (1968), 393-414.
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37-65
Gábor P. Nagy and 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
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