q-Eulerian 数的 q-log-concavity 的组合证明

Xinmiao Liu, Jiangxia Hou, Fengxia Liu
{"title":"q-Eulerian 数的 q-log-concavity 的组合证明","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&lt; k &lt;n\\)</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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&lt; k &lt;n\\\\)</span>.</p>\",\"PeriodicalId\":501430,\"journal\":{\"name\":\"The Ramanujan Journal\",\"volume\":null,\"pages\":null},\"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}","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

摘要

Carlitz 建立了欧拉数 \(A_{n,k}(q)\ 的 q-analog 并定义了 \(A_{n,k}(q)=q^{frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)\ 的关系。)本文利用 \(A_{n,k}^{*}(q)\) 的组合解释并构造注入映射,证明 \(A_{n,k}^{*}(q)\) 和 \(A_{n,k}(q)\) 都是 q-log-concave 的,也就是说,多项式 \(( A_{n,k}^{*}(q)) 的所有系数都是^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) ()和(((A_{n,k}(q))^{2}- A_{n,k-1}(q) A_{n,k+1}(q)\) 对于 \(1< k <n\) 都是非负的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

摘要图片

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
A combinatorial proof of q-log-concavity of q-Eulerian numbers

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 \(1< k <n\).

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
On the periods of twisted moments of the Kloosterman connection Ramanujan’s missing hyperelliptic inversion formula A q-analog of the Stirling–Eulerian Polynomials Integer group determinants of order 16 Diophantine approximation with prime denominator in quadratic number fields under GRH
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1