cp's OEIS Frontend

It is not clear whether the empty quandle is connected, so the sequence starts at order 1 instead of 0.

A homogeneous quandle is a quandle whose automorphism group acts transitively on the elements of the quandle.

The terms can be calculated by using the Mace4C system which is an isomorph-free model finder. - Choiwah Chow, Oct 30 2023

A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.

Also the number of isomorphism classes of Legendrian racks of order n; see Ta, "Good involutions...," Theorem 10.1.

Also the number of isomorphism classes of GL-quandles of order n; see Ta, "Classification and...," Theorem 5.6.

A quandle is Latin if its multiplication table is a Latin square. A Latin quandle may be described as a left (or right) distributive quasigroup. Sherman Stein (see reference below) proved that a left distributive quasigroup of order n exists if and only if n is not of the form 4k + 2.

A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle. Since a simple quandle with more than two elements is connected, we have a(n) <= A181771(n), for n > 2, with equality if n is prime.

Some authors consider the quandle with one element to be simple and some do not.

Since a primitive quandle is connected, we have a(n) <= A181771(n) for all n.

Furthermore, since a primitive quandle is simple, we have a(n) <= A196111(n) for all n.

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.

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; cf. Cattabriga and Nasybullov, Section 3.2.

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.