{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:17:21Z","timestamp":1759335441846,"version":"3.41.2"},"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 in the present work a recurrence relation, which has long been overlooked, for the $q$-Eulerian polynomial $A_n^{{\\rm des},{\\rm inv}}(t,q) =\\sum_{\\sigma\\in\\mathfrak{S}_n} t^{{\\rm des}(\\sigma)}q^{{\\rm inv}(\\sigma)}$, where ${\\rm des}(\\sigma)$ and ${\\rm inv}(\\sigma)$ denote, respectively, the descent number and inversion number of $\\sigma$ in the symmetric group $\\mathfrak{S}_n$ of degree $n$.  We give an algebraic proof and a combinatorial proof of the recurrence relation.<\/jats:p>","DOI":"10.37236\/471","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:13:56Z","timestamp":1578716036000},"source":"Crossref","is-referenced-by-count":3,"title":["A Recurrence Relation for the \"inv\" Analogue of $q$-Eulerian Polynomials"],"prefix":"10.37236","volume":"17","author":[{"given":"Chak-On","family":"Chow","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,4,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1n22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1n22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:54:04Z","timestamp":1579305244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1n22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/471","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,4,19]]},"article-number":"N22"}}