A177886 The number of isomorphism classes of Latin quandles (a.k.a. left distributive quasigroups) of order n.

1, 0, 1, 1, 3, 0, 5, 2, 8, 0, 9, 1, 11, 0, 5, 9, 15, 0, 17, 3, 7, 0, 21, 2, 34, 0, 62, 7, 27, 0, 29, 8, 11, 0, 15, 9, 35, 0, 13, 6, 39, 0, 41, 9, 36, 0, 45

Offset

1,5

Comments

A quandle is Latin if its multiplication table is a Latin square. A Latin quandle may be described as a left (or right) distributive quasigroup. Sherman Stein (see reference below) proved that a left distributive quasigroup of order n exists if and only if n is not of the form 4k + 2.

Links

Table of n, a(n) for n=1..47.

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.

G. Ehrman, A. Gurpinar, M. Thibault, and D. Yetter, Some Sharp Ideas on Quandle Construction

A. Hulpke, D. Stanovský, and P. Vojtěchovský, Connected quandles and transitive groups, arXiv:1409.2249 [math.GR], 2014.

S. Nelson, A polynomial invariant of finite quandles, arXiv:math/0702038 [math.QA], 2007.

S. K. Stein, On the Foundations of Quasigroups, Transactions of American Mathematical Society, 85 (1957), 228-256.

Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341 [math.GT], 2011-2012.

Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.

Index entries for quandles.

Example

a(2) = 0 since the only quandle of order 2 has multiplication table with rows [1,1] and [2,2].

Prog

(GAP) # (using the Rig package)
LoadPackage("rig");
a:=[1, 0];;
Print(1, ", ");
Print(0, ", ");
for n in [3..35] do
  a[n]:=0;
  for i in [1..NrSmallQuandles(n)] do
    if IsLatin(SmallQuandle(n, i)) then
      a[n]:=a[n]+1;
    fi;
  od;
  Print(a[n], ", ");
od; # W. Edwin Clark, Nov 26 2011

Crossrefs

Cf. A181769, A176077, A181771.

Sequence in context: A175297 A165754 A193067 * A011293 A181884 A284662

Adjacent sequences: A177883 A177884 A177885 * A177887 A177888 A177889

Keyword

nonn,more,changed

Author

W. Edwin Clark, Dec 14 2010

Extensions

Added fact due to S. K. Stein that a(4k+2) = 0 and a reference to Stein's paper.

a(11)-a(35) from W. Edwin Clark, Nov 26 2011

Links to the rig Gap package by W. Edwin Clark, Nov 26 2011

a(36)-a(47) by David Stanovsky, Oct 01 2014

Status

approved