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 Jedlička, Pilitowska, Stanovský, and Zamojska-Dzienio.