{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828790,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/6330","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:55:00Z","timestamp":1578671700000},"source":"Crossref","is-referenced-by-count":3,"title":["Counting racks of order $n$"],"prefix":"10.37236","volume":"24","author":[{"given":"Matthew","family":"Ashford","sequence":"first","affiliation":[]},{"given":"Oliver","family":"Riordan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,6,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:56:51Z","timestamp":1579237011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,2]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6330","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,6,2]]},"article-number":"P2.32"}}