Counting racks of order n

A rack on [n] can be thought of as a set of maps (fx)x∈[n], where each fx is a permutation of [n] such that f(x)fy = f −1 y fxfy for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [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)fy = z, and applying various combinatorial tools.