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The number of isomorphism classes of connected Alexander (a.k.a. indecomposable affine) quandles of order n.
2

%I #33 Dec 26 2021 23:33:54

%S 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 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 0,57,3,59,0,40,61,33,0,65,15,21,0,69,16,71,0,34

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

%C 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 W. Edwin Clark, <a href="/A193067/b193067.txt">Table of n, a(n) for n = 1..255</a>

%H W. E. Clark, M. Elhamdadi, M. Saito, 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 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 16</a>, arXiv:math/0409460 [math.GT], 2004-2006.

%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Medial">Medial</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 CA:=[];;

%o k:=8;;

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

%o CA[n]:=0;

%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 if Size(N2) = n then CA[n]:=CA[n]+1; fi;

%o od;

%o od;

%o for j in [1..k] do

%o if n = 2^j and n <> 2^(j-1) then Print("done to ",n, "\n"); fi;

%o od;

%o od;

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

%o Print(CA[n], ",");

%o od;

%Y See Index to OEIS under quandles.

%K nonn

%O 1,5

%A _W. Edwin Clark_, Jul 15 2011