ENUMERATION OF RACKS AND QUANDLES UP TO ISOMORPHISM

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders n <= 13 up to isomorphism, improving upon the previously known results for n <= 8 and n <= 9, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of order <= 11 and quandles of order <= 12. For the remaining orders we merely count the isomorphism types, relying in part on the enumeration of 2-reductive racks and 2-reductive quandles due to Jedlicka, Pilitowska, Stanovsky, and Zamojska-Dzienio.