2 June 2017
Electronic Journal of Combinatorics
© 2017, Australian National University. All rights reserved. A rack on [n] can be thought of as a set of maps (fx)x∈[n], where each fxis a permutation of [n] such that f(x)fy=f−1yfxfyfor 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))n2and at most 2(c+o(1))n2, where c≈1.557; in this paper we improve the upper bound to 2(1/4+o(1))n2, 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.
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