{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:36Z","timestamp":1753893816962,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\\rm Cat}^m(A_n)$, where $m\\in\\mathbb{N}_{&gt;0}$. In particular, the map which we define bijects facets containing the negative simple root $-\\alpha$ to dominant regions having the hyperplane $\\{v\\in V\\mid\\left\\langle v,\\alpha \\right\\rangle=m\\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\\Delta^m(A_n)$ and the set of bounded dominant regions in ${\\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.<\/jats:p>","DOI":"10.37236\/3169","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:13:19Z","timestamp":1578705199000},"source":"Crossref","is-referenced-by-count":3,"title":["Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type $A$"],"prefix":"10.37236","volume":"20","author":[{"given":"Susanna","family":"Fishel","sequence":"first","affiliation":[]},{"given":"Myrto","family":"Kallipoliti","sequence":"additional","affiliation":[]},{"given":"Eleni","family":"Tzanaki","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,10,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:12:03Z","timestamp":1579259523000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,21]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3169","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,10,21]]},"article-number":"P7"}}