{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:38Z","timestamp":1753893818739,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A rack of order $n$ is a binary operation $\\vartriangleright$ on a set $X$ of\u00a0cardinality $n$, such that right multiplication is an\u00a0automorphism. More precisely, $(X,\\vartriangleright)$ is a rack provided that\u00a0the map $x\\mapsto x\\vartriangleright y$ is a bijection for all $y\\in X$, and $(x\\vartriangleright y)\\vartriangleright z=(x\\vartriangleright z)\\vartriangleright (y\\vartriangleright z)$ for all $x,y,z\\in X$.The paper provides upper and lower bounds of the form $2^{cn^2}$ on\u00a0the number of isomorphism classes of racks of order $n$. Similar\u00a0results on the number of isomorphism classes of quandles and kei are\u00a0obtained. The results of the paper are established by first showing\u00a0how an arbitrary rack is related to its operator group (the\u00a0permutation group on $X$ generated by the maps $x\\mapsto x\\vartriangleright\u00a0y$\u00a0for $y\\in Y$), and then applying some of the theory of\u00a0permutation groups. The relationship between a rack and its operator\u00a0group extends results of Joyce and of Ryder; this relationship might\u00a0be of independent interest.<\/jats:p>","DOI":"10.37236\/3262","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:22:05Z","timestamp":1578705725000},"source":"Crossref","is-referenced-by-count":2,"title":["Enumerating Finite Racks, Quandles and Kei"],"prefix":"10.37236","volume":"20","author":[{"given":"Simon R.","family":"Blackburn","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,9,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:13:23Z","timestamp":1579259603000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,20]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,7,19]]}},"URL":"https:\/\/doi.org\/10.37236\/3262","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,9,20]]},"article-number":"P43"}}