{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T20:56:29Z","timestamp":1764276989145,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A $k \\times n$ Latin rectangle $L$ is a $k \\times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols.  If $k=n$ then $L$ is a Latin square.  Let $L_{k,n}$ be the number of $k \\times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers.  We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French.  Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$.  We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese.  We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless.  We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.<\/jats:p>","DOI":"10.37236\/487","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:11:52Z","timestamp":1578715912000},"source":"Crossref","is-referenced-by-count":22,"title":["The Many Formulae for the Number of Latin Rectangles"],"prefix":"10.37236","volume":"17","author":[{"given":"Douglas S.","family":"Stones","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,6,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1a1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1a1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:50:43Z","timestamp":1579305043000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1a1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/487","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,6,14]]},"article-number":"A1"}}