A386233 -id:A386233 - cp's OEIS frontend

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

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
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# 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.

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

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

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