{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:36Z","timestamp":1753893816471,"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>Given sets $X$ and $Y$ of positive integers and a permutation $\\sigma = \\sigma_1 \\sigma_2 \\cdots \\sigma_n \\in S_n$, an $(X,Y)$-descent of $\\sigma$ is a descent pair $\\sigma_i &gt; \\sigma_{i+1}$ whose \"top\" $\\sigma_i$ is in $X$ and whose \"bottom\" $\\sigma_{i+1}$ is in $Y$. Recently Hall and Remmel proved two formulas for the number $P_{n,s}^{X,Y}$ of $\\sigma \\in S_n$ with $s$ $(X,Y)$-descents, which generalized Liese's results in [1].  We define a new statistic ${\\rm stat}_{X,Y}(\\sigma)$ on permutations $\\sigma$ and define $P_{n,s}^{X,Y}(q)$ to be the sum of $q^{{\\rm stat}_{X,Y}(\\sigma)}$ over all $\\sigma \\in S_n$ with $s$ $(X,Y)$-descents. We then show that there are natural $q$-analogues of the Hall-Remmel formulas for $P_{n,s}^{X,Y}(q)$.<\/jats:p>","DOI":"10.37236\/200","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:25:21Z","timestamp":1578698721000},"source":"Crossref","is-referenced-by-count":1,"title":["$q$-Counting Descent Pairs with Prescribed Tops and Bottoms"],"prefix":"10.37236","volume":"16","author":[{"given":"John","family":"Hall","sequence":"first","affiliation":[]},{"given":"Jeffrey","family":"Liese","sequence":"additional","affiliation":[]},{"given":"Jeffrey B.","family":"Remmel","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,8,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r111\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r111\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T21:41:04Z","timestamp":1579297264000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r111"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/200","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,8,31]]},"article-number":"R111"}}