{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:40:55Z","timestamp":1753890055644,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2020,1,23]],"date-time":"2020-01-23T00:00:00Z","timestamp":1579737600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2020,1,23]],"date-time":"2020-01-23T00:00:00Z","timestamp":1579737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2020,1,23]],"date-time":"2020-01-23T00:00:00Z","timestamp":1579737600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,4,1]]},"abstract":"<jats:p>Linear chord diagrams are partitions of $\\left[2n\\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\\{i,i+1\\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.<\/jats:p><jats:p>Comment: 10 pages, final revisions<\/jats:p>","DOI":"10.23638\/dmtcs-21-2-11","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:30Z","timestamp":1743699390000},"source":"Crossref","is-referenced-by-count":0,"title":["Statistics on Linear Chord Diagrams"],"prefix":"10.23638","volume":"Vol. 21 no. 2, Permutation...","author":[{"given":"Naiomi T.","family":"Cameron","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kendra","family":"Killpatrick","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2020,1,23]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1902.09021v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1902.09021v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:30Z","timestamp":1743699390000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/5211"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,23]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/dmtcs-21-2-11","relation":{"has-preprint":[{"id-type":"arxiv","id":"1902.09021v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1902.09021v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1902.09021","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1902.09021","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2020,1,23]]},"article-number":"5211"}}