A181771 -id:A181771 - cp's OEIS frontend

A181769 Number of isomorphism classes of quandles of order n.

1, 1, 1, 3, 7, 22, 73, 298, 1581, 11079, 102771, 1275419, 21101335, 469250886

Offset: 0

Quandles up to order 8 were determined first by Sam Nelson and co-authors (see references). Nelson's results were confirmed independently by the submitter, and extended to order 9.

M. Elhamdadi, Distributivity in Quandles and Quasigroups, arXiv preprint arXiv:1209.6518 [math.RA], 2012. - From _N. J. A. Sloane_, Dec 29 2012
Richard Henderson, Todd Macedo and Sam Nelson, Symbolic Computation with Finite Quandles, J. Symb. Comp. 41 (2006), 811-817.
Benita Ho and Sam Nelson, Matrices and Finite Quandles, Homology, Homotopy and Applications, 7 (2005), No. 1, 197-208.
P. Jedlicka, A. Pilitowska, D. Stanovsky et al., The structure of medial quandles, arXiv preprint 1409.8396 [math.GR], 2014-2015.
J. McCarron, Connected Quandles with Order Equal to Twice an Odd Prime, arXiv preprint arXiv:1210.2150 [math.GR], 2012. - From _N. J. A. Sloane_, Dec 31 2012
Sam Nelson, Quandles and Racks
Petr Vojtěchovský and Seung Yeop Yang, Enumeration of racks and quandles up to isomorphism, Math. Comp. 88 (2019), 2523-2540; arXiv:1911.04991 [math.QA], 2019.
Wikipedia, Quandles
Index entries for sequences related to quandles and racks

Cf. A176077, A181771, A165200, A179010, A177886, A178432, A181770.

a(10)-a(13) from Petr Vojtěchovský and Seung Yeop Yang added by Andrei Zabolotskii, Jun 15 2022

A176077 Number of isomorphism classes of homogeneous quandles of order n.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 6, 15, 14, 14, 10, 61, 12, 25, 33, 142, 16, 203, 18, 266, 94, 127, 22

Offset: 1

W. Edwin Clark, Dec 06 2010

A homogeneous quandle is a quandle whose automorphism group acts transitively on the elements of the quandle.

			a(2) = 1 since for order 2 there is only the trivial quandle with product x*y=x for all x,y. The trivial quandle has automorphism group S_2 which acts transitively on the two element quandle.

David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37-65
Sam Nelson, Quandles and Racks
Wikipedia, Quandles.

Cf. A181771, A181769.

More terms from James McCarron, Aug 26 2011

A179010 The number of isomorphism classes of commutative quandles of order n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 7

Offset: 1

W. Edwin Clark, Jan 04 2011

A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.

V. D. Belousov, The structure of distributive quasigroups, (Russian) Mat. Sb. (N.S.) 50 (92) 1960 267-298.
George Glauberman, On Loops of Odd Order II, Journal of Algebra 8 (1968), 393-414.
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37-65
Gábor P. Nagy and Petr Vojtchovský, The Moufang loops of order 64 and 81, Journal of Symbolic Computation, Volume 42 Issue 9, September, 2007.
Wikipedia, Racks and quandles

Cf. A181769, A176077, A181771, A000688.

Results due to Belousov, Nagy and Vojtchovský, and Glauberman added, and sequence extended to n = 81, by W. Edwin Clark, Jan 25 2011

In Comments section, "Every commutative quandle" replaced with "Every finite commutative quandle" by W. Edwin Clark, Mar 09 2014

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

Original entry on oeis.org

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

W. Edwin Clark, Dec 14 2010

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.

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

W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013
G. Ehrman, A. Gurpinar, M. Thibault, D. Yetter, Some Sharp Ideas on Quandle Construction
A. Hulpke, D. Stanovský, 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:1105.5341v1 [math.GT].
Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.

Cf. A181769, A176077, A181771.

See also Index to OEIS under quandles.

(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

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

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

Original entry on oeis.org

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

Offset: 2

James McCarron, Oct 27 2011

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.

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

W. E. Clark, M. Elhamdadi, M. Saito, and 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

Cf. A181769, A181771.

See also Index to OEIS under quandles.

# 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)a[u]); # W. Edwin Clark, Dec 06 2011

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

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

A226173 The number of connected keis (involutory quandles) of order n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 3, 1, 0, 4, 0, 1, 3, 1, 3, 4, 0, 1, 10, 2, 0, 8, 2, 1, 10, 1, 0, 2, 0, 1, 16, 1, 0, 2, 8, 1, 8, 1, 0, 13, 0, 1

Offset: 1

W. Edwin Clark, May 29 2013

A quandle (Q,*) is a kei (also called involutory quandle) if for all x,y in Q we have (x*y)*y = x, that is, all right translations R_a: x-> x*a, are involutions.

J. S. Carter, A survey of quandle ideas. in: Kauffman, Louis H. (ed.) et al., Introductory lectures on knot theory, Series on Knots and Everything 46, World Scientific (2012), 22--53.
W. E. Clark, M. Elhamdadi, M. Saito and T. Yeatman, Quandle colorings of knots and applications. J. Knot Theory Ramifications 23/6 (2014), 1450035.

N. Andruskiewitsch, M. Graňa, From racks to pointed Hopf algebras, Adv. Math. 178/2 (2003), 177-243.
J. Scott Carter, A Survey of Quandle Ideas, arXiv:1002.4429 [math.GT], Feb 2010
W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013
A. Hulpke, D. Stanovský, P. Vojtěchovský, Connected quandles and transitive groups, arXiv:1409.2249 [math.GR], Sep 2014, to appear in J. Pure Appl. Algebra.

Cf. A181771 (number of connected quandles of order n).

See also Index to OEIS under quandles.

a(36)-a(47) (calculated by methods described in Hulpke, Stanovský, Vojtěchovský link) from David Stanovsky, Jun 02 2015

A236146 Number of primitive quandles of order n, up to isomorphism. A quandle is primitive if its inner automorphism groups acts primitively on it.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 2, 3, 1, 9, 0, 11, 1, 3, 15, 0, 17, 0, 1, 0, 21, 0, 10, 0, 8, 2, 27, 0, 29, 6, 0, 0, 0

Offset: 1

James McCarron, Feb 03 2014

Since a primitive quandle is connected, we have a(n) <= A181771(n) for all n.

Furthermore, since a primitive quandle is simple, we have a(n) <= A196111(n) for all n.

James McCarron, Table of n, a(n) for n = 1..34
Wikipedia, Racks and quandles
James McCarron, Connected Quandles with Order Equal to Twice an Odd Prime
Leandro Vendramin, Doubly transitive groups and cyclic quandles

Cf. A181771, A181769, A196111.

For odd primes p, a(p) = p - 2.

A225744 The number of isomorphism classes of connected, Generalized Alexander quandles of order n.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 3, 8, 0, 9, 3, 11, 0, 3, 9, 15, 0, 17, 3, 5, 0, 21, 5, 34, 0, 35, 5, 27, 0, 29, 17, 9, 0, 15, 18, 35, 0, 11, 9, 39, 0, 41, 9, 24, 0, 45, 21, 76, 0, 15, 11, 51, 0, 27, 19, 17, 0, 57, 15, 59, 0, 40, 97, 33, 0, 65, 15, 21, 0, 69, 37, 71, 0, 39, 17, 45, 0, 77, 34, 218, 0, 81, 15, 45, 0, 27, 27, 87, 0, 55, 21, 29, 0, 51, 43, 95, 0, 72, 34

Offset: 1

W. Edwin Clark, Aug 04 2013

nonn

Given a group G and an automorphism f of G define the binary operation * on G by x*y = f(xy^(-1))y. Then (G,*) is a quandle. We call this a Generalized Alexander quandle. If G is abelian then (G,*) is an Alexander quandle (see A193024). (G,*) is connected if the group generated by the right translations of (G,*) is transitive on G.

J. Scott Carter, A Survey of Quandle Ideas, arXiv:1002.4429 [math.GT]
W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

Cf. A193067, A181771.

See also Index to OEIS under quandles.

IsConnected:=function(A)
local B,LL;
B:=TransposedMat(A);
   LL:=List(B,x->PermList(x));
   return IsTransitive(Group(LL),[1..Length(A)]);
end;;
MakeGAlex:=function(f,g)
local e,n,QM,i,j;
  e:=Elements(g);
  n:=Length(e);
  QM:=List([1..n],t->[1..n]);
    for i in [1..n] do
      for j in [1..n] do
       QM[i][j]:=Position(e,Image(f,e[i]*e[j]^(-1))*e[j]);
      od;
    od;
  return QM;
end;;
a:=[];;
for n in [1..100] do
a[n]:=0;
N:=NrSmallGroups(n);
for u in [1..N] do
   g:=SmallGroup(n,u);
   ag:=AutomorphismGroup(g);;
   eag:=List(ConjugacyClasses(ag),Representative);
   for t in eag do
      QM:=MakeGAlex(t,g);
      if IsConnected(QM) then a[n]:=a[n]+1; fi;
   od;
  od;
od;;
a;

A226172 The number of faithful connected quandles of order n.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 2, 8, 1, 9, 8, 11, 0, 7, 9, 15, 12, 17, 9, 9, 0, 21, 27, 34, 0, 62, 13, 27, 20, 29, 8, 11, 0, 15

Offset: 1

W. Edwin Clark, May 29 2013

Let Q be a quandle with product *. Let R_a: x ->x*a for a,x in Q. By definition R_a is an automorphism of the quandle Q. If the mapping a->R_a is an injection then the quandle is said to be faithful. A quandle is faithful precisely when the columns of the Cayley table of the quandle are distinct.
This sequence was computed using Leandro Vendramin's list of all connected quandles of order at most 35.
Warning: Vendramin's quandles are assumed to be left distributive and hence one must reverse the order of the product to get the (right distributive) quandles used by knot theorists.

W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013
F. J. B. J. Clauwens, Small Connected Quandles, arxiv 1011.2456, Nov 10 2010
Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341v1 [math.GT], May 26 2011

Cf. A181771 (gives the number of connected quandles of order n).

See also Index to OEIS under quandles.

A226174 The number of self-dual connected quandles of order n.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 1, 4, 1, 1, 10, 1, 0, 5, 5, 1, 8, 1, 8, 5, 0, 1, 40, 6, 0, 21, 3, 1, 18, 1, 7, 3, 0, 1

Offset: 1

W. Edwin Clark, May 29 2013

Given a quandle (Q,*) the dual quandle is (Q,o) where c = a*b if and only if a = cob. If a quandle is isomorphic to its dual quandle it is said to be self-dual.

W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito, Timothy Yeatman, Connected Quandles Associated with Pointed Abelian Groups
W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

Cf. A181771 (number of connected quandles of order n).

See also Index to OEIS under quandles.

cp's OEIS Frontend

A181769 Number of isomorphism classes of quandles of order n.

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A176077 Number of isomorphism classes of homogeneous quandles of order n.

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A179010 The number of isomorphism classes of commutative quandles of order n.

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

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GAP

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

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A226173 The number of connected keis (involutory quandles) of order n.

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A236146 Number of primitive quandles of order n, up to isomorphism. A quandle is primitive if its inner automorphism groups acts primitively on it.

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A225744 The number of isomorphism classes of connected, Generalized Alexander quandles of order n.

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GAP

A226172 The number of faithful connected quandles of order n.

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