{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817342,"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>Let $\\mathcal{C}_n$ be the set of Dyck paths of length $n$. In this paper, by a new automorphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set $\\mathcal{C}_n$ is equidistributed with the statistic 'number of up steps at height $h$ with $h\\equiv 0$ (mod 3)'. Moreover, for $m\\ge 3$, we prove that the two statistics 'number of up steps at height $h$ with $h\\equiv 0$ (mod $m$)' and 'number of up steps at height $h$ with $h\\equiv m-1$ (mod $m$)' on the set $\\mathcal{C}_n$ are 'almost equidistributed'.  Both results are proved combinatorially.<\/jats:p>","DOI":"10.37236\/579","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:48:25Z","timestamp":1578696505000},"source":"Crossref","is-referenced-by-count":0,"title":["Exterior Pairs and Up Step Statistics on Dyck Paths"],"prefix":"10.37236","volume":"18","author":[{"given":"Sen-Peng","family":"Eu","sequence":"first","affiliation":[]},{"given":"Tung-Shan","family":"Fu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,4,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p92\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:13:04Z","timestamp":1579284784000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,21]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/579","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,4,21]]},"article-number":"P92"}}