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A177886
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The number of isomorphism classes of latin quandles of order n.
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1
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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
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OFFSET
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1,5
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COMMENTS
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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.
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REFERENCES
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S. K. Stein, On the Foundations of Quasigroups, Transactions of American Mathematical Society, 85 (1957),228-256.
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LINKS
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Table of n, a(n) for n=1..35.
G. Ehrman, A. Gurpinar, M. Thibault, D. Yetter, Some Sharp Ideas on Quandle Construction
S. Nelson, A polynomial invariant of finite quandles
S. K. Stein, On the foundations of quasigroups
Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341v1 [math.GT].
Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.
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EXAMPLE
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a(2) = 0 since the only quandle of order 2 has multiplication table with rows [1,1] and [2,2].
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PROG
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GAP with 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;
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CROSSREFS
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Cf. A181769, A176077, A181771
Sequence in context: A175297 A165754 A193067 * A011293 A181884 A201565
Adjacent sequences: A177883 A177884 A177885 * A177887 A177888 A177889
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KEYWORD
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nonn,more
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AUTHOR
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W. Edwin Clark, Dec 14 2010
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EXTENSIONS
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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
GAP program to compute the sequence up to n = 35 by W. Edwin Clark, Nov 26 2011
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STATUS
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approved
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