A178432 -id:A178432 - cp's OEIS frontend

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

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
```

A386234 Number of good involutions of all nontrivial core quandles of order n.

Original entry on oeis.org

1, 4, 1, 3, 1, 72, 2, 3, 1, 31, 1, 3, 1, 10856, 1, 7, 1, 47, 2, 3, 1

Offset: 3

Luc Ta, Jul 21 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.

A core quandle Core(G) is a group G viewed as a kei (i.e., involutory quandle) under the operation g(h) = g*h^-1*g. Note that Core(G) is nontrivial if and only if exp(G) > 2.

			For n = 4 the only nontrivial core quandle is the dihedral quandle R4 = Core(Z/4Z) of order 4. It is well-known (see Thm. 3.2 of Kamada and Oshiro) that R4 has exactly four good involutions. Hence a(4) = 4.
For n = 6 the only nontrivial core quandles are Core(S3) and R6 = Core(Z/6Z), which have one and two good involutions, respectively. Hence a(6) = 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, 101-108.

Seiichi Kamada and Kanako Oshiro, Homology groups of symmetric quandles and cocycle invariants of links and surface-links, Trans. Amer. Math. Soc., 362 (2010), no. 10, 5501-5527.
Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 3.
Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.
Index entries for sequences related to quandles and racks

Cf. A000001, A386233, A181770, A178432, A386231, A386232.

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

Let n > 2. Then Ta, Cor. 7.17 implies the following. If n appears in A000040 or A050384, then a(n) = 1. If n appears in A221048, then a(n) = 2. If n > 4 and n appears in A100484, then a(n) = 3.

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

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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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A386234 Number of good involutions of all nontrivial core quandles of order n.

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

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

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