{"title":"A group action on cyclic compositions and γ-positivity","authors":"Shishuo Fu , Jie Yang","doi":"10.1016/j.ejc.2024.104107","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> be the number of Dyck paths of semilength <span><math><mi>n</mi></math></span> with <span><math><mi>k</mi></math></span> occurrences of <span><math><mrow><mi>U</mi><mi>D</mi></mrow></math></span> and <span><math><mi>m</mi></math></span> occurrences of <span><math><mrow><mi>U</mi><mi>U</mi><mi>D</mi></mrow></math></span>. We establish in two ways a new interpretation of the numbers <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, 2-dominant compositions) is introduced, which amounts to give a combinatorial proof of the <span><math><mi>γ</mi></math></span>-positivity of the Narayana polynomial, as well as the <span><math><mi>γ</mi></math></span>-positivity of the polynomial <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>∑</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>k</mi></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub><msup><mrow><mi>t</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></math></span> previously obtained by Bóna et al, with apparently new combinatorial interpretations of their <span><math><mi>γ</mi></math></span>-coefficients.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104107"},"PeriodicalIF":1.0000,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001926","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be the number of Dyck paths of semilength with occurrences of and occurrences of . We establish in two ways a new interpretation of the numbers in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, 2-dominant compositions) is introduced, which amounts to give a combinatorial proof of the -positivity of the Narayana polynomial, as well as the -positivity of the polynomial previously obtained by Bóna et al, with apparently new combinatorial interpretations of their -coefficients.
期刊介绍:
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.