cp's OEIS Frontend

A196111 Number of isomorphism classes of simple quandles of order n.

%I A196111 #41 Feb 13 2025 18:01:02
%S A196111 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 A196111 0,0,0,3,35,0,0,2,39,3,41,0,3,0,45
%N A196111 Number of isomorphism classes of simple quandles of order n.
%C A196111 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 A196111 Some authors consider the quandle with one element to be simple and some do not.
%H A196111 W. E. Clark, M. Elhamdadi, M. Saito, and T. Yeatman, <a href="http://arxiv.org/abs/1312.3307">Quandle Colorings of Knots and Applications</a>, arXiv preprint arXiv:1312.3307, 2013
%H A196111 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 A196111 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 A196111 Leandro Vendramin and Matías Graña, <a href="http://code.google.com/p/rig/">Rig, a GAP package for racks and quandles</a>.
%H A196111 Wikipedia, <a href="http://en.wikipedia.org/wiki/Racks_and_quandles">Racks and quandles</a>
%F A196111 a(p) = A181771(p) = p - 2, for prime p > 2.
%e A196111 a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).
%o A196111 (GAP) # Using the Rig package.
%o A196111 LoadPackage("rig");
%o A196111 IsSimpleQuandle:=function(q)
%o A196111 local g,N,gg,n;
%o A196111 if IsFaithful(q) = false then return false; fi;
%o A196111 g:=InnerGroup(q);;
%o A196111 if Size(Center(g))>1 then return false; fi;
%o A196111 N:=NormalSubgroups(g);;
%o A196111 gg:=DerivedSubgroup(g);;
%o A196111 for n in N do
%o A196111   if Size(n) = 1 then continue; fi;
%o A196111   if IsSubset(gg,n) and Size(n)<Size(gg) then return false;  fi;
%o A196111 od;
%o A196111 return true;
%o A196111 end;;
%o A196111 a:=[1,1];;
%o A196111 for n in [3..35] do
%o A196111   a[n]:=0;
%o A196111   for i in [1..NrSmallQuandles(n)] do
%o A196111     if IsSimpleQuandle(SmallQuandle(n,i)) then
%o A196111       a[n]:=a[n]+1;
%o A196111     fi;
%o A196111   od;
%o A196111 od;
%o A196111 List([1..35],u->a[u]); # _W. Edwin Clark_, Dec 06 2011
%Y A196111 Cf. A181769, A181771.
%Y A196111 See also Index to OEIS under quandles.
%K A196111 nonn,hard,more
%O A196111 2,4
%A A196111 _James McCarron_, Oct 27 2011
%E A196111 a(21) corrected by _W. Edwin Clark_, Dec 06 2011
%E A196111 a(32)-a(35) added by _W. Edwin Clark_, Dec 06 2011
%E A196111 a(36)-a(47) added by _W. Edwin Clark_, Dec 28 2014