{"paper":{"title":"Enumerating finite racks, quandles and kei","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.GT","authors_text":"Simon R. Blackburn","submitted_at":"2012-03-29T12:19:33Z","abstract_excerpt":"A rack of order $n$ is a binary operation $\\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\\rack)$ is a rack provided that the map $x\\mapsto x\\rack y$ is a bijection for all $y\\in X$, and $(x\\rack y)\\rack z=(x\\rack z)\\rack (y\\rack z)$ for all $x,y,z\\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6504","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}