{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:46:08Z","timestamp":1772448368707,"version":"3.50.1"},"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>Each positive rational number $x&gt;0$ can be written uniquely as $x=a\/(b-a)$ for coprime positive integers $0&lt;a&lt;b$. We will identify $x$ with the pair $(a,b)$. In this paper we define for each positive rational $x&gt;0$ a simplicial complex $\\mathsf{Ass}(x)=\\mathsf{Ass}(a,b)$ called the\u00a0rational associahedron. \u00a0It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the rational Catalan number\u00a0$$\\mathsf{Cat}(x)=\\mathsf{Cat}(a,b):=\\frac{(a+b-1)!}{a!\\,b!}.$$The cases $(a,b)=(n,n+1)$ and\u00a0$(a,b)=(n,kn+1)$ recover the classical associahedron and its \"Fuss-Catalan\"\u00a0generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. \u00a0We prove that $\\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the rational Narayana numbers) and $f$-vector (the rational Kirkman numbers). \u00a0We define $\\mathsf{Ass}(a,b)$ via rational Dyck paths: lattice paths from $(0,0)$ to $(b,a)$ staying above the line\u00a0$y = \\frac{a}{b}x$. \u00a0We also use rational Dyck paths to define a rational generalization\u00a0of noncrossing perfect matchings of $[2n]$. \u00a0In the case $(a,b) = (n, mn+1)$, our construction produces the noncrossing partitions\u00a0of $[(m+1)n]$ in which each block has size $m+1$.<\/jats:p>","DOI":"10.37236\/3432","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:21:39Z","timestamp":1578705699000},"source":"Crossref","is-referenced-by-count":20,"title":["Rational Associahedra and Noncrossing Partitions"],"prefix":"10.37236","volume":"20","author":[{"given":"Drew","family":"Armstrong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brendon","family":"Rhoades","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nathan","family":"Williams","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2013,9,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p54\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:12:30Z","timestamp":1579259550000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,26]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,7,19]]}},"URL":"https:\/\/doi.org\/10.37236\/3432","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,9,26]]},"article-number":"P54"}}