{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:39:59Z","timestamp":1775468399377,"version":"3.50.1"},"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>Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and\u00a0double quotients $W \/ W_I$ and $W_I \\backslash W \/ W_J$, appear in many contexts in\u00a0combinatorics and Lie theory, including the geometry and topology of\u00a0generalized flag varieties and the symmetry groups of regular polytopes. The\u00a0set of ordinary cosets $w W_I$, for $I \\subseteq S$, forms the Coxeter complex of $W$,\u00a0and is well-studied. In this article we look at a less studied object: the set of\u00a0all double cosets $W_I w W_J$ for $I, J \\subseteq S$.\u00a0Double cosets are not uniquely presented by triples\u00a0$(I,w,J)$. We describe what we call the lex-minimal presentation, and\u00a0prove that there exists a unique such object for each double coset.\u00a0Lex-minimal presentations are then used to enumerate double cosets via a finite automaton depending on the Coxeter graph for $(W,S)$.\u00a0As an example, we present a formula for the number of parabolic double\u00a0cosets with a fixed minimal element when $W$ is the symmetric group\u00a0$S_n$ (in this case, parabolic subgroups are also known as Young\u00a0subgroups). Our formula is almost always linear time computable in\u00a0$n$, and we show how it can be generalized to any Coxeter\u00a0group with little additional work. We spell out formulas for all finite and\u00a0affine Weyl groups in the case that $w$ is the identity element.<\/jats:p>","DOI":"10.37236\/6741","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:40:52Z","timestamp":1578670852000},"source":"Crossref","is-referenced-by-count":6,"title":["Parabolic Double Cosets in Coxeter Groups"],"prefix":"10.37236","volume":"25","author":[{"given":"Sara C.","family":"Billey","sequence":"first","affiliation":[]},{"given":"Matja\u017e","family":"Konvalinka","sequence":"additional","affiliation":[]},{"given":"T. Kyle","family":"Petersen","sequence":"additional","affiliation":[]},{"given":"William","family":"Slofstra","sequence":"additional","affiliation":[]},{"given":"Bridget E.","family":"Tenner","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,2,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p23\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i1p23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:38:57Z","timestamp":1579235937000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i1p23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/6741","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,16]]},"article-number":"P1.23"}}