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

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

Offset

0,3

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

Table of n, a(n) for n=0..8.

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

Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Prog

(GAP) # See Ta, GitHub link

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.

Sequence in context: A149169 A138544 A219287 * A359922 A093149 A048054

Adjacent sequences: A386228 A386229 A386230 * A386232 A386233 A386234

Keyword

nonn,hard,more

Author

Luc Ta, Jul 16 2025

Status

approved