James McCarron - cp's OEIS frontend

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.

A198147 Numbers k for which there are non-isomorphic groups of order k whose core quandles are isomorphic.

Original entry on oeis.org

27, 32, 54, 64, 81, 96, 100, 108, 125, 128, 135, 147, 160, 162, 189, 192, 200, 216, 224, 243, 250, 260, 270, 273, 288, 294, 297, 300, 320, 324, 340, 343, 351, 352, 375, 378, 399, 400, 405, 416, 432, 441, 448, 459, 480, 486, 500, 507, 513, 520, 540, 546, 567, 580, 588

Offset: 1

James McCarron, Dec 21 2012

This is a subsequence of sequence A118642.

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

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.

A180043 The number of isomorphism classes of Szasz (uniquely non-associative) groupoids of order n.

Original entry on oeis.org

0, 0, 10, 24, 2064, 39961, 1194828

Offset: 1

James McCarron, Jan 14 2011

A Szasz groupoid (S,*) is one for which there is exactly one ordered triple (a,b,c) of members of S that does not associate: (a*b)*c != a*(b*c). For any other triple (x,y,z), we have (x*y)*z = x*(y*z). Thus, a Szasz groupoid is as close to being a semigroup as possible, without actually being associative. G. Szasz proved that such groupoids exist on any set with at least four members. Every Szasz groupoid is non-commutative.

			The "smallest" Szasz groupoid of order 3 with elements {a,b,c} defines c*b = b, and the product of every other pair of elements is defined to be a. Then, (c*c)*b = a*b = b but c*(c*b) = c*b = b, but every triple other than (c,c,b) associates.

G. Szasz, Die Unabhangigkeit der Assoziativitatsbedingungen, Acta. Sci. Math. Szeged 15 (1953), 20-28.

A178432 Number of isomorphism classes of kei (involutory quandles) of order n.

Original entry on oeis.org

1, 1, 1, 3, 5, 13, 41, 142, 665, 4288, 36455, 436672, 6926801

Offset: 0

James McCarron, Dec 21 2010

The terms can be calculated by using the Mace4C system which is an isomorph-free model finder. - Choiwah Chow, Oct 30 2023

P. Jedlicka, A. Pilitowska, D. Stanovsky et al., The structure of medial quandles, arXiv preprint 1409.8396 [math.GR], 2014.
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37-65.
Sam Nelson, Quandles and Racks
Mituhisa Takasaki, Abstraction of symmetric transformations (also referenced as Abstractions of symmetric functions), Tohoku Math. J., 49 (1943), 143-207 [in Japanese].

Cf. A181769, A226173, A242044.

a(11)-a(12) from Choiwah Chow, Oct 30 2023

A181771 Number of isomorphism classes of connected quandles of order n.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 5, 3, 8, 1, 9, 10, 11, 0, 7, 9, 15, 12, 17, 10, 9, 0, 21, 42, 34, 0, 65, 13, 27, 24, 29, 17, 11, 0, 15, 73, 35, 0, 13, 33, 39, 26, 41, 9, 45, 0, 45

Offset: 1

James McCarron

It is not clear whether the empty quandle is connected, so the sequence starts at order 1 instead of 0.

Hulpke, A. Personal communication, 2014.
Holt, D.; Royle, G. Personal communication, 2014.

John J. Cannon and Derek F. Holt, The transitive permutation groups of degree 32, Experiment. Math. 17 (2008), no. 3, 307--314.
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
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.
Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341 [math.GT], 2011-2012.
Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.

Cf. A181769, A176077.

# (using the Rig package)
LoadPackage("rig");
for n in [1..47] do  Display(NrSmallQuandles(n));  od;
# Leandro Vendramin, Sep 14 2014

Ninth term corrected by James McCarron, Dec 05 2010

More terms from Leandro Vendramin, Sep 14 2014

A181770 Number of isomorphism classes of racks of order n.

Original entry on oeis.org

1, 1, 2, 6, 19, 74, 353, 2080, 16023, 159526, 2093244, 36265070, 836395102, 25794670618

Offset: 0

James McCarron

Also the number of isomorphism classes of Legendrian racks of order n; see Ta, "Good involutions...," Theorem 10.1.

Also the number of isomorphism classes of GL-quandles of order n; see Ta, "Classification and...," Theorem 5.6.

Přemysl Jedlička, Agata Pilitowska, and Anna Zamojska-Dzienio, Multipermutation set theoretic solutions of the Yang-Baxter equation of level 2, arXiv:1901.01471 [math.QA], 2019.
Sam Nelson, Quandles and Racks
Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025.
Lực Ta, Classification and structure of generalized Legendrian racks, arXiv:2504.12671 [math.GT], 2025.
Petr Vojtěchovský and Seung Yeop Yang, Enumeration of racks and quandles up to isomorphism, Mathematics of Computation, 88 (2019), 2523-2540; arXiv:1911.04991 [math.QA], 2019.

Cf. A181769, A383144-A383146, A383828-A383831, A385040, A385041, A386231-A386234.

a(9)-a(13) from Petr Vojtěchovský and Seung Yeop Yang added by Andrey Zabolotskiy, Jun 15 2022

A181769 Number of isomorphism classes of quandles of order n.

Original entry on oeis.org

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

Offset: 0

James McCarron

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

A165200 Number of isomorphism classes of abelian / medial quandles.

Original entry on oeis.org

1, 1, 1, 3, 6, 18, 58, 251, 1410, 10311, 98577, 1246488, 20837439, 466087635

Offset: 0

James McCarron, Jan 12 2011

A quandle is abelian / medial (both names are being used) if it satisfies the identity (XY)(UV) = (XU)(YV). Not to be confused with a commutative quandle (A179010).

P. Jedlička, A. Pilitowska, D. Stanovský, A. Zamojska-Dzienio, The structure of medial quandles, arXiv preprint 1409.8396 [math.GR], 2014.
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37-65.
Sam Nelson, Quandles and Racks
David Stanovský, Calculating with quandles, GAP code to calculate the numbers.
Wikipedia, Quandles
Wikipedia, Medial magma

Cf. A179010 (commutative quandles), A242044, A242275.

More terms from David Stanovsky, Sep 30 2014

Description edited by W. Edwin Clark, May 30 2013, and David Stanovsky, Sep 30 2014

cp's OEIS Frontend

User: James McCarron

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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A198147 Numbers k for which there are non-isomorphic groups of order k whose core quandles are isomorphic.

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

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A180043 The number of isomorphism classes of Szasz (uniquely non-associative) groupoids of order n.

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A178432 Number of isomorphism classes of kei (involutory quandles) of order n.

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A181771 Number of isomorphism classes of connected quandles of order n.

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A181770 Number of isomorphism classes of racks of order n.

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A181769 Number of isomorphism classes of quandles of order n.

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A165200 Number of isomorphism classes of abelian / medial quandles.

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