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Number of isomorphism classes of simple quandles of order n.
1

%I #36 Dec 26 2021 21:08:43

%S 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,

%T 0,0,0,3,35,0,0,2,39,3,41,0,3,0,45

%N Number of isomorphism classes of simple quandles of order n.

%C 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.

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

%H W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307, 2013

%H David Joyce, <a href="http://dx.doi.org/10.1016/0021-8693(82)90305-2">Simple Quandles</a>, J. Algebra 79(2) 1982, 307-318.

%H Leandro Vendramin, <a href="http://arxiv.org/abs/1105.5341">On the classification of quandles of low order</a>, arXiv:1105.5341v1 [math.GT].

%H Leandro Vendramin and Matías Graña, <a href="http://code.google.com/p/rig/">Rig, a GAP package for racks and quandles</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Racks_and_quandles">Racks and quandles</a>

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

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

%o (GAP) (using the Rig package)

%o LoadPackage("rig");

%o IsSimpleQuandle:=function(q)

%o local g,N,gg,n;

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

%o g:=InnerGroup(q);;

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

%o N:=NormalSubgroups(g);;

%o gg:=DerivedSubgroup(g);;

%o for n in N do

%o if Size(n) = 1 then continue; fi;

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

%o od;

%o return true;

%o end;;

%o a:=[1,1];;

%o for n in [3..35] do

%o a[n]:=0;

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

%o if IsSimpleQuandle(SmallQuandle(n,i)) then

%o a[n]:=a[n]+1;

%o fi;

%o od;

%o od;

%o List([1..35],u->a[u]); # _W. Edwin Clark_, Dec 06 2011

%Y Cf. A181769, A181771.

%Y See also Index to OEIS under quandles.

%K nonn,hard,more

%O 2,4

%A _James McCarron_, Oct 27 2011

%E a(21) corrected by _W. Edwin Clark_, Dec 06 2011

%E a(32)-a(35) added by _W. Edwin Clark_, Dec 06 2011

%E a(36)-a(47) added by _W. Edwin Clark_, Dec 28 2014