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

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

Offset: 1

Author

W. Edwin Clark, Jul 15 2011

Keywords

• nonn

Comments

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

Links

• W. Edwin Clark, Table of n, a(n) for n = 1..255
• W. Edwin Clark, M. Elhamdadi, M. Saito, and T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307 [math.GT], 2013-2014.
• S. Nelson, Classification of Finite Alexander Quandles, arXiv:math/0202281 [math.GT], 2002-2003.
• S. Nelson, Alexander Quandles of Order 16, arXiv:math/0409460 [math.GT], 2004-2006.
• K. Toyoda, On axioms of linear functions, Proceedings of the Imperial Academy 17/7(1941), 221-227.
• Wikipedia, Medial magma
• Index entries for sequences related to quandles and racks

Programs

GAP

findY:=function(f,g)
local Y,y;
  Y:=[];
  for y in g do
    Add(Y,Image(f,y^(-1))*y);
  od;
  Y:=Set(Y);
  return Subgroup(g,Y);
end;;
CA:=[];;
k:=8;;
for n in [1..2^k-1] do
   CA[n]:=0;
   LGn:=AllSmallGroups(n,IsAbelian);
   for g in LGn do
     autg:=AutomorphismGroup(g);;
     eautg:=List(ConjugacyClasses(autg),Representative);
     for f in eautg do
       N2:=findY(f,g);
       if Size(N2) = n then CA[n]:=CA[n]+1; fi;
     od;
   od;
   for j in [1..k] do
   if n = 2^j and n <> 2^(j-1) then Print("done to ",n, "\n"); fi;
   od;
od;
for n in [1..2^k-1] do
  Print(CA[n], ",");
od;