{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836374,"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>This paper deals with different computational methods to enumerate the set $\\mathrm{PLR}(r,s,n;m)$ of $r \\times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \\leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\\mathrm{PLR}(r,s,n;m)$, for all $r \\leqslant s \\leqslant n \\leqslant 7$, and all $r \\leqslant s \\leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \\leqslant s \\leqslant n \\leqslant 6$. Numerical results have been cross-checked where possible.<\/jats:p>","DOI":"10.37236\/9093","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:52:16Z","timestamp":1593132736000},"source":"Crossref","is-referenced-by-count":3,"title":["Enumerating Partial Latin Rectangles"],"prefix":"10.37236","volume":"27","author":[{"given":"Ra\u00fal M.","family":"Falc\u00f3n","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rebecca J.","family":"Stones","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,6,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p47\/8104","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p47\/8104","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:52:32Z","timestamp":1593132752000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,12]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/9093","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,6,12]]},"article-number":"P2.47"}}