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

1, 1, 2, 5, 12, 38, 168, 850, 6090

Offset: 0

Author

Luc Ta, May 11 2025

Keywords

• nonn
• hard
• more

Comments

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.

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, pages 101-108.

Links

• 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.

Crossrefs

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.

Programs

GAP

# See Ta, GitHub link