Counting racks of order n

Author: 

Ashford, M
Riordan, O

Publication Date: 

2 June 2017

Journal: 

Electronic Journal of Combinatorics

Last Updated: 

2020-01-21T09:07:10.903+00:00

Issue: 

2

Volume: 

24

abstract: 

© 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.

Symplectic id: 

636614

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article