{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:26Z","timestamp":1753893806932,"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>Let ${\\cal A}\\subseteq {\\bf [n]}\\times{\\bf [n]}$ be a set of pairs containing the diagonal ${\\cal D} = \\{(i,i)\\,|\\, i=1,\\ldots,n\\}$, and such that $a\\leq b$ for all $(a,b) \\in {\\cal A}$. We study formulae for the generating series $F_{\\cal A} ({\\bf x}) = \\sum_w {\\bf x}^w$ where the sum is over all words $w \\in {\\bf [n]}^*$ that avoid ${\\cal A}$, i.e., $(w_i,w_{i+1})\\notin {\\cal A}$ for $i=1,\\ldots,|w|-1$. This series is a rational function, with denominator of the form $1-\\sum_{T}\\mu_{{\\cal A}}(T){\\bf x}^T$, where the sum is over all nonempty subsets $T$ of $[n]$. Our principal focus is the case where the relation ${\\cal A}$ is $\\mu$-positive, i.e., $\\mu_{\\cal A}(T)\\ge 0$ for all $T\\subseteq {\\bf [n]}$, in which case the form of the generating function suggests a cancellation-free combinatorial encoding of words avoiding ${\\cal A}$. We supply such an interpretation for several classes of examples, including the interesting class of cycle-free (or crown-free) posets.<\/jats:p>","DOI":"10.37236\/1885","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:16:51Z","timestamp":1578719811000},"source":"Crossref","is-referenced-by-count":2,"title":["Words Avoiding a Reflexive Acyclic Relation"],"prefix":"10.37236","volume":"11","author":[{"given":"John","family":"Dollhopf","sequence":"first","affiliation":[]},{"given":"Ian","family":"Goulden","sequence":"additional","affiliation":[]},{"given":"Curtis","family":"Greene","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2006,2,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:26:16Z","timestamp":1579321576000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i2r28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,2,8]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2004,6,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1885","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2006,2,8]]},"article-number":"R28"}}