{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:52Z","timestamp":1753893832374,"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>Given a finite poset $P$, we associate a simple graph denoted by $G_P$ with all connected order ideals of $P$ as vertices, and two vertices are adjacent if and only if they have nonempty intersection and are incomparable with respect to set inclusion. We establish a bijection between the set of maximum independent sets of $G_P$ and the set of $P$-forests, introduced by Feray and Reiner in their study of the fundamental generating function $F_P(\\textbf{x})$ associated with $P$-partitions. Based on this bijection, in the cases when $P$ is naturally labeled we show that $F_P(\\textbf{x})$ can factorise, such that each factor is a summation of rational functions determined by maximum independent sets of a connected component of $G_P$. This approach enables us to give an alternative proof for Feray and Reiner's nice formula of $F_P(\\textbf{x})$ for the case of $P$ being a naturally labeled forest with duplications. Another consequence of our result is a product formula to compute the number of linear extensions of $P$.<\/jats:p>","DOI":"10.37236\/6463","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:36:17Z","timestamp":1578670577000},"source":"Crossref","is-referenced-by-count":0,"title":["Connected Order Ideals and $P$-Partitions"],"prefix":"10.37236","volume":"25","author":[{"given":"Ben P.","family":"Zhou","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,3,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p65\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p65\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:37:32Z","timestamp":1579235852000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i1p65"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/6463","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,3,16]]},"article-number":"P1.65"}}