Counting racks of order n

A rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , where c≈1.557; in this paper we improve the upper bound to 2 (1/4+o(1)) n 2 , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on [n], where we have an edge of colour y between x and z if and only if (x)f y =z, and applying various combinatorial tools.