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

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

Offset: 0

Author

Luc Ta, Jul 16 2025

Keywords

• nonn
• hard
• more

Comments

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.

References

• 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.

Links

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

Crossrefs

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.

Programs

GAP

# See Ta, GitHub link