A383828 Number of involutory racks of order n, up to isomorphism.

1, 1, 2, 5, 13, 42, 180, 906, 6317

Offset

0,3

Comments

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.

a(n) is also the number of Legendrian kei (i.e., kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.

a(n) is also the number of symmetric kei (i.e., 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

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

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.

Prog

(GAP) # See Ta, GitHub link

Crossrefs

Cf. A383829-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.

Sequence in context: A178682 A229161 A192745 * A299430 A307263 A278171

Adjacent sequences: A383825 A383826 A383827 * A383829 A383830 A383831

Keyword

nonn,hard,more

Author

Luc Ta, May 11 2025

Status

approved