login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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

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 267-298. George Glauberman, On

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, 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 20 03:47 EDT 2020. Contains 337264 sequences. (Running on oeis4.)