{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:05Z","timestamp":1753893845596,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>\u00a0Using the standard Coxeter presentation for the symmetric group $\\mathfrak{S}_{n}$, two reduced expressions for the same group element $\\textsf{w}$ are said to be commutationally equivalent if one expression can be obtained from the other one\u00a0 by applying a finite sequence of commutations. The commutation classes can be seen as the vertices of a graph $\\widehat{G}(\\textsf{w})$, where two classes are connected by an edge if elements of those classes differ by a long braid relation. We compute the radius and diameter\u00a0of the graph $\\widehat{G}(\\textsf{w}_{\\bf 0})$,\u00a0 for the longest element\u00a0 $\\textsf{w}_{\\bf 0}$ in the symmetric group $\\mathfrak{S}_{n}$, and show that it is not a planar graph for $n\\geq 6$. We also describe a family of commutation classes which contains all atoms, that is classes with one single element, and a subfamily of commutation classes whose elements are in bijection with standard Young tableaux of certain moon-polyomino shapes.<\/jats:p>","DOI":"10.37236\/9481","type":"journal-article","created":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T04:52:28Z","timestamp":1589518348000},"source":"Crossref","is-referenced-by-count":1,"title":["Commutation Classes of the Reduced Words for the Longest Element of ${\\mathfrak S}_n$"],"prefix":"10.37236","volume":"27","author":[{"given":"Gon\u00e7alo","family":"Gutierres","sequence":"first","affiliation":[]},{"given":"Ricardo","family":"Mamede","sequence":"additional","affiliation":[]},{"given":"Jos\u00e9 Luis","family":"Santos","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,5,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p21\/8078","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p21\/8078","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T04:52:28Z","timestamp":1589518348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,15]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/9481","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,15]]},"article-number":"P2.21"}}