{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T03:46:18Z","timestamp":1773891978631,"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 the joint distribution of descents and inverse descents over the set of permutations of $n$ letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture.\u00a0 We examine generalizations of the conjecture and obtain a type $B$ analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.<\/jats:p>","DOI":"10.37236\/2135","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:57:06Z","timestamp":1578711426000},"source":"Crossref","is-referenced-by-count":6,"title":["Some Remarks on the Joint Distribution of Descents and Inverse Descents"],"prefix":"10.37236","volume":"20","author":[{"given":"Mirk\u00f3","family":"Visontai","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,3,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p52\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:25:22Z","timestamp":1579260322000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i1p52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,8]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/2135","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,8]]},"article-number":"P52"}}