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

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, 18, 28, 8, 30, 48, 20, 16, 24, 33, 36, 18, 24, 28, 40, 12, 42, 30, 44, 22, 46, 46, 83, 39, 32, 36, 52, 45, 40, 42, 36, 28, 58, 24, 60, 30, 66, 167, 48, 20, 66, 48

Offset: 1

Author

W. Edwin Clark, Jul 15 2011

Keywords

• nonn

Comments

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).

Links

• W. Edwin Clark, Table of n, a(n) for n = 1..255
• W. E. Clark, M. Elhamdadi, M. Saito and T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307 [math.GT], 2013-2014.
• M. Elhamdadi, Distributivity in Quandles and Quasigroups, arXiv preprint arXiv:1209.6518 [math.RA], 2012. - From _N. J. A. Sloane_, Dec 29 2012
• Xiang-dong Hou, Finite Modules over Z[t,t^{-1}], arXiv:1107.2076 [math.RA], 2011.
• S. Nelson, Classification of Finite Alexander Quandles, arXiv:math/0202281 [math.GT], 2002-2003.
• S. Nelson, Alexander Quandles of Order 16s, arXiv:math/0409460 [math.GT], 2004-2006.
• Wikipedia, Racks and Quandles

Crossrefs

See Index to OEIS under quandles.

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;;
Alex:=[];;k:=8;;
for nn in [1..2^k-1] do
  Alex[nn]:=0;
od;
for n in [1..2^k-1] do
  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);
      MM:= ((Size(g)^2)/Size(N2));
      for nn in [1..2^k-1] do
        if nn mod MM = 0 then
          Alex[nn]:=Alex[nn]+1;
        fi;
      od;
    od;
  od;
od;
for nn in [1..2^k-1] do
  Print(Alex[nn], ",");
od;;