{"paper":{"title":"Facets of the $m$-generalized cluster complex and regions in the $m$-extended Catalan arrangement of type $A_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eleni Tzanaki, Myrto Kallipoliti, Susanna Fishel","submitted_at":"2013-02-11T10:48:45Z","abstract_excerpt":"In this paper we present a bijection $\\omega_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\\Delta^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\\rm Cat}^m(A_n)$, where $m\\in\\mathbb{N}_{>0}$. In particular, $\\omega_n$ bijects the facets containing the negative simple root $-\\alpha$ to dominant regions having the hyperplane $\\{v\\in V\\mid<v,\\alpha >=m\\}$ as separating wall. As a result, $\\omega_n$ restricts to a bijection between the set of facets of the positive part of $\\Delta^m(A_n)$ and the set of bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2441","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}