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A193067 The number of isomorphism classes of connected Alexander (a.k.a. indecomposable affine) quandles of order n.

%I A193067 #42 Aug 06 2025 15:05:43
%S A193067 1,0,1,1,3,0,5,2,8,0,9,1,11,0,3,9,15,0,17,3,5,0,21,2,34,0,30,5,27,0,
%T A193067 29,8,9,0,15,8,35,0,11,6,39,0,41,9,24,0,45,9,76,0,15,11,51,0,27,10,17,
%U A193067 0,57,3,59,0,40,61,33,0,65,15,21,0,69,16,71,0,34
%N A193067 The number of isomorphism classes of connected Alexander (a.k.a. indecomposable affine) quandles of order n.
%C A193067 Finite connected Alexander (affine) quandles are Latin. According to the Toyoda-Bruck theorem, Latin affine quandles are the same objects as idempotent medial quasigroups. The values up to 16 were obtained by Nelson (see links below). - Edited by _David Stanovsky_, Oct 01 2014
%H A193067 W. Edwin Clark, <a href="/A193067/b193067.txt">Table of n, a(n) for n = 1..255</a>
%H A193067 W. Edwin 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 A193067 S. Nelson, <a href="http://arxiv.org/abs/math/0202281">Classification of Finite Alexander Quandles</a>, arXiv:math/0202281 [math.GT], 2002-2003.
%H A193067 S. Nelson, <a href="http://arxiv.org/abs/math/0409460">Alexander Quandles of Order 16</a>, arXiv:math/0409460 [math.GT], 2004-2006.
%H A193067 K. Toyoda, <a href="http://dx.doi.org/10.3792/pia/1195578751">On axioms of linear functions</a>, Proceedings of the Imperial Academy 17/7(1941), 221-227.
%H A193067 Wikipedia, <a href="https://en.wikipedia.org/wiki/Medial_magma">Medial magma</a>
%H A193067 <a href="/index/Qua#quandles">Index entries for sequences related to quandles and racks</a>
%o A193067 (GAP)
%o A193067 findY:=function(f,g)
%o A193067 local Y,y;
%o A193067   Y:=[];
%o A193067   for y in g do
%o A193067     Add(Y,Image(f,y^(-1))*y);
%o A193067   od;
%o A193067   Y:=Set(Y);
%o A193067   return Subgroup(g,Y);
%o A193067 end;;
%o A193067 CA:=[];;
%o A193067 k:=8;;
%o A193067 for n in [1..2^k-1] do
%o A193067    CA[n]:=0;
%o A193067    LGn:=AllSmallGroups(n,IsAbelian);
%o A193067    for g in LGn do
%o A193067      autg:=AutomorphismGroup(g);;
%o A193067      eautg:=List(ConjugacyClasses(autg),Representative);
%o A193067      for f in eautg do
%o A193067        N2:=findY(f,g);
%o A193067        if Size(N2) = n then CA[n]:=CA[n]+1; fi;
%o A193067      od;
%o A193067    od;
%o A193067    for j in [1..k] do
%o A193067    if n = 2^j and n <> 2^(j-1) then Print("done to ",n, "\n"); fi;
%o A193067    od;
%o A193067 od;
%o A193067 for n in [1..2^k-1] do
%o A193067   Print(CA[n], ",");
%o A193067 od;
%K A193067 nonn
%O A193067 1,5
%A A193067 _W. Edwin Clark_, Jul 15 2011