{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:30Z","timestamp":1753893870620,"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>In 2010, Bousquet-M\u00e9lou et al. defined sequences of\u00a0nonnegative integers called ascent sequences and showed that\u00a0the ascent sequences of length $n$ are in one-to-one correspondence\u00a0with the interval orders, i.e., the posets not containing the poset\u00a0$\\mathbf{2}+\\mathbf{2}$. Through the use of generating\u00a0functions, this provided an answer to the longstanding open question\u00a0of enumerating the (unlabeled) interval orders. A semiorder is an\u00a0interval order having a representation in which all intervals have\u00a0the same length. In terms of forbidden subposets, the semiorders\u00a0exclude $\\mathbf{2}+\\mathbf{2}$ and $\\mathbf{1}+\\mathbf{3}$. The number of unlabeled\u00a0semiorders on $n$ points has long been known to be the\u00a0$n$th Catalan number. However, describing the\u00a0ascent sequences that correspond to the semiorders under the\u00a0bijection of Bousquet-M\u00e9lou et al. has proved difficult. In this\u00a0paper, we discuss a major part of the difficulty in this area: the\u00a0ascent sequence corresponding to a semiorder may have an initial\u00a0subsequence that corresponds to an interval order that is not a\u00a0semiorder.\r\nWe define the hereditary semiorders to be those corresponding to an\u00a0ascent sequence for which every initial subsequence also corresponds\u00a0to a semiorder. We provide a structural result that characterizes\u00a0the hereditary semiorders and use this characterization to determine\u00a0the ordinary generating function for hereditary semiorders. We also\u00a0use our characterization of hereditary semiorders and the\u00a0characterization of semiorders of dimension $3$ given by Rabinovitch\u00a0to provide a structural description of the semiorders of dimension\u00a0at most $2$. From this description, we are able to determine the\u00a0ordinary generating function for the semiorders of dimension at most $2$.<\/jats:p>","DOI":"10.37236\/8140","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:37:36Z","timestamp":1584578256000},"source":"Crossref","is-referenced-by-count":0,"title":["Hereditary Semiorders and Enumeration of Semiorders by Dimension"],"prefix":"10.37236","volume":"27","author":[{"given":"Mitchel T.","family":"Keller","sequence":"first","affiliation":[]},{"given":"Stephen J.","family":"Young","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,3,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p50\/8040","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p50\/8040","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:37:36Z","timestamp":1584578256000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8140","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,3,6]]},"article-number":"P1.50"}}