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Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of size<jats:italic>k<\/jats:italic>are in correspondence with vertices of out-degree<jats:italic>k<\/jats:italic>. This allows us to obtain limit theorems concerning the block structure of simply generated non-crossing partitions. We apply our results in free probability by giving a simple formula relating the maximum of the support of a compactly supported probability measure on the real line in terms of its free cumulants.<\/jats:p>","DOI":"10.1017\/s0963548317000050","type":"journal-article","created":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T09:49:18Z","timestamp":1490694558000},"page":"560-592","source":"Crossref","is-referenced-by-count":9,"title":["Simply Generated Non-Crossing Partitions"],"prefix":"10.1017","volume":"26","author":[{"given":"IGOR","family":"KORTCHEMSKI","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CYRIL","family":"MARZOUK","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,3,28]]},"reference":[{"key":"S0963548317000050_ref36","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90255-8"},{"key":"S0963548317000050_ref28","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(72)90041-6"},{"key":"S0963548317000050_ref9","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316962"},{"key":"S0963548317000050_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpb.2004.09.005"},{"key":"S0963548317000050_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v17-1773"},{"key":"S0963548317000050_ref26","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOP871"},{"key":"S0963548317000050_ref23","doi-asserted-by":"publisher","DOI":"10.1214\/11-PS188"},{"key":"S0963548317000050_ref11","first-page":"377","article-title":"Excursion normalis\u00e9e, m\u00e9andre et pont pour les processus de L\u00e9vy stables","volume":"121","author":"Chaumont","year":"1997","journal-title":"Bull. 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