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A193024 The number of isomorphism classes of Alexander (aka affine) quandles of order n. 2
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 (list; graph; refs; listen; history; text; internal format)
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, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

M. Elhamdadi, Distributivity in Quandles and Quasigroups, arXiv preprint arXiv:1209.6518, 2012. - From N. J. A. Sloane, Dec 29 2012

Xiang-dong Hou, Finite Modules over  Z[t,t^{-1}]

S. Nelson, Classification of Finite Alexander Quandles

S. Nelson, Alexander Quandles of Order 16s

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] doPrint(Alex[nn], ", "); od;;

CROSSREFS

See Index to OEIS under quandles.

Sequence in context: A178970 A172054 A047994 * A153038 A324911 A220335

Adjacent sequences:  A193021 A193022 A193023 * A193025 A193026 A193027

KEYWORD

nonn

AUTHOR

W. Edwin Clark, Jul 15 2011

STATUS

approved

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Last modified May 30 23:02 EDT 2020. Contains 334747 sequences. (Running on oeis4.)