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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.

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.

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.