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

1,3

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

Prog

(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;;

Crossrefs

See Index to OEIS under quandles.

Sequence in context: A344005 A345044 A047994 * A340368 A367175 A153038

Adjacent sequences: A193021 A193022 A193023 * A193025 A193026 A193027

Keyword

nonn

Author

W. Edwin Clark, Jul 15 2011

Status

approved