A383144 Number of abelian/medial racks of order n, up to isomorphism.

1, 1, 2, 6, 18, 68, 329, 1965, 15455, 155902, 2064870, 35982366, 832699635, 25731050872

Offset: 0

Author

Luc Ta, Apr 17 2025

Keywords

• nonn
• hard
• more

Comments

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

a(n) is also the number of medial Legendrian racks of order n up to isomorphism; see Ta, "Equivalences of...," Theorem 1.1.

a(n) is also the number of medial generalized Legendrian quandles (also called GL-quandles or bi-Legendrian quandles) of order n up to isomorphism; see Ta, "Generalized Legendrian...," Theorem 5.5.

Links

• Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv:2505.08090 [math.GT], 2025.
• Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, 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), no. 319, 2523-2540.

Crossrefs

Sequences related to medial racks and quandles: A165200, A242044, A226193, A242275, A243931, A257351, A383146, A383829, A383831.

Other sequences related to racks and quandles: A181769, A181770, A181771, A176077, A178432, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A198147, A225744, A226172, A226174, A383145.