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A193024 The number of isomorphism classes of Alexander (a.k.a. affine) quandles of order n.

%I A193024 #37 Dec 27 2021 13:15:40
%S A193024 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 A193024 18,28,8,30,48,20,16,24,33,36,18,24,28,40,12,42,30,44,22,46,46,83,39,
%U A193024 32,36,52,45,40,42,36,28,58,24,60,30,66,167,48,20,66,48
%N A193024 The number of isomorphism classes of Alexander (a.k.a. affine) quandles of order n.
%C A193024 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 A193024 W. Edwin Clark, <a href="/A193024/b193024.txt">Table of n, a(n) for n = 1..255</a>
%H A193024 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 A193024 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 A193024 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 A193024 S. Nelson, <a href="http://arxiv.org/abs/math/0202281">Classification of Finite Alexander Quandles</a>, arXiv:math/0202281 [math.GT], 2002-2003.
%H A193024 S. Nelson, <a href="http://arxiv.org/abs/math/0409460">Alexander Quandles of Order 16s</a>, arXiv:math/0409460 [math.GT], 2004-2006.
%H A193024 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quandle">Racks and Quandles </a>
%o A193024 (GAP)
%o A193024 findY:=function(f,g)
%o A193024 local Y,y;
%o A193024   Y:=[];
%o A193024   for y in g do
%o A193024     Add(Y,Image(f,y^(-1))*y);
%o A193024   od;
%o A193024   Y:=Set(Y);
%o A193024   return Subgroup(g,Y);
%o A193024 end;;
%o A193024 Alex:=[];;k:=8;;
%o A193024 for nn in [1..2^k-1] do
%o A193024   Alex[nn]:=0;
%o A193024 od;
%o A193024 for n in [1..2^k-1] do
%o A193024   LGn:=AllSmallGroups(n,IsAbelian);
%o A193024   for g in LGn do
%o A193024     autg:=AutomorphismGroup(g);;
%o A193024     eautg:=List(ConjugacyClasses(autg),Representative);
%o A193024     for f in eautg do
%o A193024       N2:=findY(f,g);
%o A193024       MM:= ((Size(g)^2)/Size(N2));
%o A193024       for nn in [1..2^k-1] do
%o A193024         if nn mod MM = 0 then
%o A193024           Alex[nn]:=Alex[nn]+1;
%o A193024         fi;
%o A193024       od;
%o A193024     od;
%o A193024   od;
%o A193024 od;
%o A193024 for nn in [1..2^k-1] do
%o A193024   Print(Alex[nn], ",");
%o A193024 od;;
%Y A193024 See Index to OEIS under quandles.
%K A193024 nonn
%O A193024 1,3
%A A193024 _W. Edwin Clark_, Jul 15 2011