A383828 -id:A383828 - cp's OEIS frontend

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

A383831 Number of medial Legendrian quandles of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 14, 48, 219, 1207, 9042

Offset: 0

Luc Ta, May 16 2025

A rack or quandle is medial if it satisfies the identity (xy)(uv) = (xu)(yv).
A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.
a(n) is also the number of medial racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson, Legendrian rack invariants of Legendrian knots, Communications of the Korean Mathematical Society, 36 (2021), no. 3, 623-639.
Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv:2505.08090 [math.GT], 2025.
Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv:2504.12671 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383828-A383830, A383145, A383146, A181769, A181770, A178432.
Sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# See Ta, GitHub link
```

A385041 Number of isomorphism classes of virtual quandles of order n.

Original entry on oeis.org

1, 1, 2, 8, 26, 104, 467, 2540, 18419

Offset: 0

Luc Ta, Jun 16 2025

A virtual quandle is a quandle equipped with a distinguished quandle automorphism. Two virtual quandles (Q1,f1), (Q2,f2) are isomorphic if there exists a quandle isomorphism g: Q1 -> Q2 such that g*f1 = f2*g.

Alessia Cattabriga and Timur Nasybullov, Virtual quandle for links in lens spaces, RACSAM, 112 (2008), no. 3, 657-669.
Vassily Olegovich Manturov, On invariants of virtual links, Acta Applicandae Mathematicae, 72 (2002), no. 3, 295-309.
Lực Ta, Virtual-Rack Classification, GitHub, 2025.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 5.

Cf. A385040, A383145, A181769, A181770.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432

GAP
```
See Ta, GitHub link
```

a(4)-a(8) corrected by Luc Ta, Jul 05 2025

A386231 Number of symmetric racks of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 4, 9, 42, 154, 1064, 6678, 73780

Offset: 0

Luc Ta, Jul 16 2025

A good involution f of a rack R is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (R,f) a symmetric rack. A symmetric rack isomorphism is a rack isomorphism that intertwines good involutions.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 1.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Cf. A386232, A181770, A383828, A386233, A386234.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432, A385041, A383145, A181769.

GAP
```
# See Ta, GitHub link
```

A386233 Number of good involutions of all nontrivial conjugation quandles of order A060652(n).

Original entry on oeis.org

1, 32, 1, 17, 1, 13056, 66, 33, 1, 1

Offset: 1

Luc Ta, Jul 16 2025

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle.

A conjugation quandle is a group viewed as a quandle under the conjugation operation. Since conjugation quandles of abelian groups are trivial, this sequence only considers nonabelian groups.

			For n = 1, 3, 5, 9, 10, there is a unique nonabelian group G of order A060652(n), and G is centerless. It follows from Ta, Prop. 5.3 that a(n) = 1.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 2.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Cf. A000001, A060652, A060689, A386233, A181769, A383828, A386231, A386232.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432, A385041, A383145, A181770.

GAP
```
See Ta, GitHub link
```

A383829 Number of medial involutory racks of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 12, 38, 168, 850, 6090

Offset: 0

Luc Ta, May 11 2025

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.
A rack is medial if it satisfies the identity (xy)(uv) = (xu)(yv).
a(n) is also the number of medial Legendrian kei (i.e., medial kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.
a(n) is also the number of medial symmetric kei (i.e., medial kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, pages 101-108.

Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson, Legendrian rack invariants of Legendrian knots, Communications of the Korean Mathematical Society, 36 (2021), no. 3, 623-639.
Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv: 2505.08090 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383828, A383830, A383831, A383145, A383146, A181769, A181770, A178432.
Sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# See Ta, GitHub link
```

A386232 Number of symmetric quandles of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 13, 44, 187, 937, 6459

Offset: 0

Luc Ta, Jul 16 2025

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle. A symmetric quandle isomorphism is a quandle isomorphism that intertwines good involutions.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 1.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Cf. A386231, A181769, A383828, A386233, A386234.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432, A385041, A383145, A181770.

GAP
```
See Ta, GitHub link
```

A383830 Number of Legendrian quandles of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 15, 54, 240, 1306, 9477

Offset: 0

Luc Ta, May 11 2025

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson, Legendrian rack invariants of Legendrian knots, Communications of the Korean Mathematical Society, 36 (2021), no. 3, 623-639.
Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv: 2505.08090 [math.GT], 2025.
Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv: 2504.12671 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383828, A383829, A383831, A383145, A383146, A181769, A181770, A178432.
Sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# See Ta, GitHub link
```

A385040 Number of isomorphism classes of virtual racks of order n.

Original entry on oeis.org

1, 1, 4, 15, 71, 350, 2372, 18543, 199491

Offset: 0

Luc Ta, Jun 16 2025

A virtual rack is a rack equipped with a distinguished rack automorphism. Two virtual racks (R1,f1), (R2,f2) are isomorphic if there exists a rack isomorphism g: R1 -> R2 such that g*f1 = f2*g.

Alessia Cattabriga and Timur Nasybullov, Virtual quandle for links in lens spaces, RACSAM, 112 (2008), no. 3, 657-669.
Vassily Olegovich Manturov, On invariants of virtual links, Acta Applicandae Mathematicae, 72 (2002), no. 3, 295-309.
Lực Ta, Virtual-Rack Classification, GitHub, 2025.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 5.

Cf. A385041, A383145, A181769, A181770.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432

GAP
```
See Ta, GitHub link
```

a(4)-a(8) corrected by Luc Ta, Jul 05 2025

cp's OEIS Frontend

A181770 Number of isomorphism classes of racks of order n.

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A383831 Number of medial Legendrian quandles of order n, up to isomorphism.

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A385041 Number of isomorphism classes of virtual quandles of order n.

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A386231 Number of symmetric racks of order n, up to isomorphism.

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A386233 Number of good involutions of all nontrivial conjugation quandles of order A060652(n).

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A383829 Number of medial involutory racks of order n, up to isomorphism.

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A386232 Number of symmetric quandles of order n, up to isomorphism.

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A383830 Number of Legendrian quandles of order n, up to isomorphism.

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GAP

A385040 Number of isomorphism classes of virtual racks of order n.

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