A386234 -id:A386234 - 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

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
```

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
```

A387317 Number of good involutions of all nontrivial linear quandles of order n.

Original entry on oeis.org

1, 4, 1, 2, 1, 44, 1, 2, 1, 414, 1, 2, 31, 5784, 1, 2, 1, 97358, 237, 2, 1, 1917064, 1, 2, 1, 42406158, 1

Offset: 3

Luc Ta, Aug 26 2025

A linear quandle is a pair (Z/nZ, k) where k is a unit in Z/nZ, viewed as an Alexander quandle under the operation a(b) := ka + (1-k)b. A linear quandle is trivial if and only if k = 1.

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). The pair (Q,f) is called a symmetric quandle.

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 twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025. See Table 2.
Lực Ta, Symmetric-Linear-Quandles, GitHub, 2025.
Index entries for sequences related to quandles and racks

Cf. A060594, A202828, A386233, A386234.

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

If A060594(n) = 2, then a(n) = 1 if n is odd, a(n) = 4 if n = 4, and a(n) = 2 otherwise. See Ta, Ex. 5.8 and Prop. 5.9.

For all n >= 1, we have a(4n) >= A202828(n), with equality if and only if n = 1. See Ta, Thm. 5.11.

Some terms corrected by Luc Ta, Sep 03 2025

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A181770 Number of isomorphism classes of racks of order n.

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A386231 Number of symmetric 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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A387317 Number of good involutions of all nontrivial linear quandles of order n.

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