{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:50Z","timestamp":1753893830371,"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>We consider the asymptotic minimum density $f(s,k)$ of monotone $k$-subwords of words over a totally ordered alphabet of size $s$. The unrestricted alphabet case, $f(\\infty,k)$, is well-studied, known for $f(\\infty,3)$ and $f(\\infty,4)$, and, in particular, conjectured to be rational for all $k$. Here we determine $f(2,k)$ for all $k$ and determine $f(3,3)$, which is already irrational. We describe an explicit construction for all $s$ which is conjectured to yield $f(s,3)$. Using our construction and flag algebra, we determine $f(4,3),f(5,3),f(6,3)$ up to $10^{-3}$ yet argue that flag algebra, regardless of computational power, cannot determine $f(5,3)$ precisely. Finally, we prove that for every fixed $k \\ge 3$, the gap between $f(s,k)$ and $f(\\infty,k)$ is $\\Theta(\\frac{1}{s})$.<\/jats:p>","DOI":"10.37236\/13478","type":"journal-article","created":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T15:16:11Z","timestamp":1743088571000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Minimum Density of Monotone Subwords"],"prefix":"10.37236","volume":"32","author":[{"given":"Raphael","family":"Yuster","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,3,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,27]],"date-time":"2025-03-27T15:16:12Z","timestamp":1743088572000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i1p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1,17]]}},"URL":"https:\/\/doi.org\/10.37236\/13478","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,3,28]]},"article-number":"P1.47"}}