A193024 - OEIS

%I #37 Dec 27 2021 13:15:40

%S 1,1,2,3,4,2,6,7,11,4,10,6,12,6,8,23,16,11,18,12,12,10,22,14,39,12,45,

%T 18,28,8,30,48,20,16,24,33,36,18,24,28,40,12,42,30,44,22,46,46,83,39,

%U 32,36,52,45,40,42,36,28,58,24,60,30,66,167,48,20,66,48

%N The number of isomorphism classes of Alexander (a.k.a. affine) quandles of order n.

%C Nelson enumerated Alexander quandles to order 16 (see the links below). The values of a(n) for n from 1 to 255 were obtained via a GAP program using ideas from Hou (see the link below).

%H W. Edwin Clark, <a href="/A193024/b193024.txt">Table of n, a(n) for n = 1..255</a>

%H W. E. Clark, M. Elhamdadi, M. Saito and T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307 [math.GT], 2013-2014.

%H M. Elhamdadi, <a href="http://arxiv.org/abs/1209.6518">Distributivity in Quandles and Quasigroups</a>, arXiv preprint arXiv:1209.6518 [math.RA], 2012. - From _N. J. A. Sloane_, Dec 29 2012

%H Xiang-dong Hou, <a href="http://arxiv.org/abs/1107.2076">Finite Modules over  Z[t,t^{-1}]</a>, arXiv:1107.2076 [math.RA], 2011.

%H S. Nelson, <a href="http://arxiv.org/abs/math/0202281">Classification of Finite Alexander Quandles</a>, arXiv:math/0202281 [math.GT], 2002-2003.

%H S. Nelson, <a href="http://arxiv.org/abs/math/0409460">Alexander Quandles of Order 16s</a>, arXiv:math/0409460 [math.GT], 2004-2006.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Quandle">Racks and Quandles </a>

%o (GAP)

%o findY:=function(f,g)

%o local Y,y;

%o   Y:=[];

%o   for y in g do

%o     Add(Y,Image(f,y^(-1))*y);

%o   od;

%o   Y:=Set(Y);

%o   return Subgroup(g,Y);

%o end;;

%o Alex:=[];;k:=8;;

%o for nn in [1..2^k-1] do

%o   Alex[nn]:=0;

%o od;

%o for n in [1..2^k-1] do

%o   LGn:=AllSmallGroups(n,IsAbelian);

%o   for g in LGn do

%o     autg:=AutomorphismGroup(g);;

%o     eautg:=List(ConjugacyClasses(autg),Representative);

%o     for f in eautg do

%o       N2:=findY(f,g);

%o       MM:= ((Size(g)^2)/Size(N2));

%o       for nn in [1..2^k-1] do

%o         if nn mod MM = 0 then

%o           Alex[nn]:=Alex[nn]+1;

%o         fi;

%o       od;

%o     od;

%o   od;

%o od;

%o for nn in [1..2^k-1] do

%o   Print(Alex[nn], ",");

%o od;;

%Y See Index to OEIS under quandles.

%K nonn

%O 1,3

%A _W. Edwin Clark_, Jul 15 2011