{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:16:19Z","timestamp":1759335379099,"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>The $(q,r)$-Eulerian polynomials are the (maj-exc,fix,exc) enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials,\u00a0Chung and Graham proved two symmetrical $q$-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic (inv-lec,pix,lec). We also prove a new recurrence formula for the $(q,r)$-Eulerian polynomials and study a $q$-analogue of Chung and Graham's restricted descent polynomials. In particular, we obtain a generalized symmetrical identity for these restricted $q$-Eulerian polynomials with a combinatorial proof.<\/jats:p>","DOI":"10.37236\/2927","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:57:14Z","timestamp":1578693434000},"source":"Crossref","is-referenced-by-count":5,"title":["On some Generalized $q$-Eulerian Polynomials"],"prefix":"10.37236","volume":"20","author":[{"given":"Zhicong","family":"Lin","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\/v20i1p55\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:25:30Z","timestamp":1579242330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i1p55"}},"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\/2927","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,3,8]]},"article-number":"P55"}}