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A196111 Number of isomorphism classes of simple quandles of order n. 1
1, 1, 1, 3, 0, 5, 2, 3, 1, 9, 1, 11, 0, 2, 3, 15, 0, 17, 2, 2, 0, 21, 1, 10, 0, 8, 2, 27, 1, 29, 6, 0, 0, 0, 3, 35, 0, 0, 2, 39, 3, 41, 0, 3, 0, 45 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle.  Since a simple quandle with more than two elements is connected, we have a(n) <= A181771(n), for n > 2, with equality if n is prime.

Some authors consider the quandle with one element to be simple and some do not.

LINKS

Table of n, a(n) for n=2..47.

W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

David Joyce, Simple Quandles, J. Algebra 79(2) 1982, 307-318.

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.

Wikipedia, Racks and quandles

FORMULA

a(p) = A181771(p) = p - 2, for prime p > 2.

EXAMPLE

a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).

PROG

(GAP) (using the Rig package)

LoadPackage("rig");

IsSimpleQuandle:=function(q)

local g, N, gg, n;

if IsFaithful(q) = false then return false; fi;

g:=InnerGroup(q);;

if Size(Center(g))>1 then return false; fi;

N:=NormalSubgroups(g);;

gg:=DerivedSubgroup(g);;

for n in N do

  if Size(n) = 1 then continue; fi;

  if IsSubset(gg, n) and Size(n)<Size(gg) then return false;  fi;

od;

return true;

end;;

a:=[1, 1];;

for n in [3..35] do

a[n]:=0;

for i in [1..NrSmallQuandles(n)] do

  if IsSimpleQuandle(SmallQuandle(n, i)) then

    a[n]:=a[n]+1;

  fi;

od;

od;

List([1..35], u->a[u]);

-W. Edwin Clark

CROSSREFS

Cf. A181769, A181771.

See also Index to OEIS under quandles.

Sequence in context: A130054 A187886 A236146 * A261628 A007431 A215447

Adjacent sequences:  A196108 A196109 A196110 * A196112 A196113 A196114

KEYWORD

nonn,hard,more

AUTHOR

James McCarron, Oct 27 2011

EXTENSIONS

a(21) corrected by W. Edwin Clark, Dec 06 2011

a(32)-a(35) added by W. Edwin Clark, Dec 06 2011

a(36)-a(47) added by W. Edwin Clark, Dec 28 2014

STATUS

approved

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Last modified October 23 21:59 EDT 2017. Contains 293814 sequences.