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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.

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle. A symmetric quandle isomorphism is a quandle isomorphism that intertwines good involutions.

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

A virtual rack is a rack equipped with a distinguished rack automorphism. Two virtual racks (R1,f1), (R2,f2) are isomorphic if there exists a rack isomorphism g: R1 -> R2 such that g*f1 = f2*g.