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

1,5

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. E. Clark, M. Elhamdadi, M. Saito, 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

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;;
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;

Crossrefs

See Index to OEIS under quandles.

Sequence in context: A098496 A175297 A165754 * A177886 A011293 A181884

Adjacent sequences: A193064 A193065 A193066 * A193068 A193069 A193070

Keyword

nonn

Author

W. Edwin Clark, Jul 15 2011

Status

approved