{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:37:52Z","timestamp":1773250672980,"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>We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order.  The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index.  This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval.  This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are \"smaller\" than polytopes.  More precisely, we conjecture that if one fixes the lengths of $x$ and $y$, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals $[x,y]$.  We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes.  As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i. e. simplices).<\/jats:p>","DOI":"10.37236\/1827","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:31:35Z","timestamp":1578709895000},"source":"Crossref","is-referenced-by-count":10,"title":["The cd-index of Bruhat Intervals"],"prefix":"10.37236","volume":"11","author":[{"given":"Nathan","family":"Reading","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,10,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r74\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r74\/comment","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r74\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:53:18Z","timestamp":1579323198000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r74"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,10,18]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1827","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,10,18]]},"article-number":"R74"}}