A rack on $[n]$ can be thought of as a set of maps $(f_x)_{x \in [n]}$, where each $f_x$ is a permutation of $[n]$ such that $f_{(x)f_y} = f_y^{-1}f_xf_y$ 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 $c \approx 1.557$; 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.
@article{10_37236_6330,
author = {Matthew Ashford and Oliver Riordan},
title = {Counting racks of order \(n\)},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {2},
doi = {10.37236/6330},
zbl = {1364.05035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6330/}
}
TY - JOUR
AU - Matthew Ashford
AU - Oliver Riordan
TI - Counting racks of order \(n\)
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6330/
DO - 10.37236/6330
ID - 10_37236_6330
ER -
%0 Journal Article
%A Matthew Ashford
%A Oliver Riordan
%T Counting racks of order \(n\)
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6330/
%R 10.37236/6330
%F 10_37236_6330
Matthew Ashford; Oliver Riordan. Counting racks of order \(n\). The electronic journal of combinatorics, Tome 24 (2017) no. 2. doi: 10.37236/6330
