{"title":"纳拉亚纳多项式和欧拉多项式的正相关性","authors":"Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh","doi":"10.1016/j.aam.2023.102656","DOIUrl":null,"url":null,"abstract":"<div><p><span><span>Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and </span>number theory. Sagan and Tirrell (2020) </span><span>[34]</span> stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly <em>γ</em>-positive polynomials from <em>γ</em>-positive polynomials. We then prove the alternating <em>γ</em>-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types <em>A</em> and <em>B</em>. In particular, by introducing the definition of colored <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span><span> Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type </span><em>B</em>. Finally, we present several identities involving the Eulerian polynomials of types <em>A</em> and <em>B</em>.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Positivity of Narayana polynomials and Eulerian polynomials\",\"authors\":\"Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh\",\"doi\":\"10.1016/j.aam.2023.102656\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span><span>Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and </span>number theory. Sagan and Tirrell (2020) </span><span>[34]</span> stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly <em>γ</em>-positive polynomials from <em>γ</em>-positive polynomials. We then prove the alternating <em>γ</em>-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types <em>A</em> and <em>B</em>. In particular, by introducing the definition of colored <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span><span> Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type </span><em>B</em>. Finally, we present several identities involving the Eulerian polynomials of types <em>A</em> and <em>B</em>.</p></div>\",\"PeriodicalId\":50877,\"journal\":{\"name\":\"Advances in Applied Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-12-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0196885823001744\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885823001744","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
伽玛正多项式经常出现在有限几何、代数组合学和数论中。Sagan 和 Tirrell(2020)[34] 偶然发现了一些单模态序列,结果发现它们是交替γ-正多项式,而不是γ-正多项式。受这项工作的启发,我们首先证明可以从γ-正多项式推导出交替γ-正多项式。然后,我们证明了与 A 型和 B 型 Narayana 多项式相关的几个多项式的交替 γ 正性和 Hurwitz 稳定性。特别是,通过引入彩色 2×n Young 图的定义,我们为与 B 型 Narayana 数相关的三个等式提供了组合解释。
Positivity of Narayana polynomials and Eulerian polynomials
Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ-positive polynomials from γ-positive polynomials. We then prove the alternating γ-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B.
期刊介绍:
Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas.
Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.