{"title":"A combinatorial proof of q-log-concavity of q-Eulerian numbers","authors":"Xinmiao Liu, Jiangxia Hou, Fengxia Liu","doi":"10.1007/s11139-024-00841-6","DOIUrl":null,"url":null,"abstract":"<p>Carlitz established a <i>q</i>-analog of the Eulerian numbers <span>\\(A_{n,k}(q)\\)</span> and defined the relationship <span>\\(A_{n,k}(q)=q^{\\frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)\\)</span>. In this paper, by using the combinatorial interpretation of <span>\\(A_{n,k}^{*}(q)\\)</span> and constructing injective maps, we prove that <span>\\(A_{n,k}^{*}(q)\\)</span> and <span>\\(A_{n,k}(q)\\)</span> are <i>q</i>-log-concave, that is, all the coefficients of the polynomials <span>\\(( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) \\)</span> and <span>\\((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q)\\)</span> are nonnegative for <span>\\(1< k <n\\)</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"45 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00841-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Carlitz established a q-analog of the Eulerian numbers \(A_{n,k}(q)\) and defined the relationship \(A_{n,k}(q)=q^{\frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)\). In this paper, by using the combinatorial interpretation of \(A_{n,k}^{*}(q)\) and constructing injective maps, we prove that \(A_{n,k}^{*}(q)\) and \(A_{n,k}(q)\) are q-log-concave, that is, all the coefficients of the polynomials \(( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) \) and \((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q)\) are nonnegative for \(1< k <n\).