Counting racks of order n
Matthew Ashford, Oliver Riordan

TL;DR
This paper establishes that the number of isomorphism classes of racks on a set of size n grows approximately as 2^{n^2/4}, matching the known lower bound and tightening the upper bound.
Contribution
It improves the upper bound on the count of racks on [n] to match the known lower bound, using combinatorial graph methods.
Findings
Number of racks on [n] is approximately 2^{n^2/4}.
Upper bound on isomorphism classes of racks is tightened to match the lower bound.
Graph-theoretic approach is used to analyze racks as edge-coloured directed multigraphs.
Abstract
A rack on can be thought of as a set of maps , where each is a permutation of such that for all and . In 2013, Blackburn showed that the number of isomorphism classes of racks on is at least and at most , where ; in this paper we improve the upper bound to , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on , where we have an edge of colour between and if and only if , and applying various combinatorial tools.
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