{"title":"1/k-欧拉多项式双γ正性的组合","authors":"Sherry H.F. Yan , Xubo Yang , Zhicong Lin","doi":"10.1016/j.jcta.2025.106092","DOIUrl":null,"url":null,"abstract":"<div><div>The <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> were introduced as ascent polynomials over <em>k</em>-inversion sequences by Savage and Viswanathan. The bi-<em>γ</em>-positivity of the <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> was known but to give a combinatorial interpretation of the corresponding bi-<em>γ</em>-coefficients still remains open. The study of the theme of bi-<em>γ</em>-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-<em>γ</em>-coefficients of <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:<ul><li><span>•</span><span><div>construct a bijection between <em>k</em>-Stirling permutations and certain forests that are named increasing pruned even <em>k</em>-ary forests;</div></span></li><li><span>•</span><span><div>introduce a generalized Foata–Strehl action on increasing pruned even <em>k</em>-ary trees which implies the longest ascent-plateau polynomials over <em>k</em>-Stirling permutations with initial letter 1 are <em>γ</em>-positive, a result that may have independent interest;</div></span></li><li><span>•</span><span><div>develop two crucial transformations on increasing pruned even <em>k</em>-ary forests to conclude our combinatorial interpretation.</div></span></li></ul></div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106092"},"PeriodicalIF":1.2000,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Combinatorics on bi-γ-positivity of 1/k-Eulerian polynomials\",\"authors\":\"Sherry H.F. Yan , Xubo Yang , Zhicong Lin\",\"doi\":\"10.1016/j.jcta.2025.106092\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> were introduced as ascent polynomials over <em>k</em>-inversion sequences by Savage and Viswanathan. The bi-<em>γ</em>-positivity of the <span><math><mn>1</mn><mo>/</mo><mi>k</mi></math></span>-Eulerian polynomials <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> was known but to give a combinatorial interpretation of the corresponding bi-<em>γ</em>-coefficients still remains open. The study of the theme of bi-<em>γ</em>-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-<em>γ</em>-coefficients of <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:<ul><li><span>•</span><span><div>construct a bijection between <em>k</em>-Stirling permutations and certain forests that are named increasing pruned even <em>k</em>-ary forests;</div></span></li><li><span>•</span><span><div>introduce a generalized Foata–Strehl action on increasing pruned even <em>k</em>-ary trees which implies the longest ascent-plateau polynomials over <em>k</em>-Stirling permutations with initial letter 1 are <em>γ</em>-positive, a result that may have independent interest;</div></span></li><li><span>•</span><span><div>develop two crucial transformations on increasing pruned even <em>k</em>-ary forests to conclude our combinatorial interpretation.</div></span></li></ul></div></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":\"217 \",\"pages\":\"Article 106092\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2025-07-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316525000871\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316525000871","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Combinatorics on bi-γ-positivity of 1/k-Eulerian polynomials
The -Eulerian polynomials were introduced as ascent polynomials over k-inversion sequences by Savage and Viswanathan. The bi-γ-positivity of the -Eulerian polynomials was known but to give a combinatorial interpretation of the corresponding bi-γ-coefficients still remains open. The study of the theme of bi-γ-positivity from a purely combinatorial aspect was proposed by Athanasiadis. In this paper, we provide a combinatorial interpretation for the bi-γ-coefficients of by using the model of certain ordered labeled forests. Our combinatorial approach consists of three main steps:
•
construct a bijection between k-Stirling permutations and certain forests that are named increasing pruned even k-ary forests;
•
introduce a generalized Foata–Strehl action on increasing pruned even k-ary trees which implies the longest ascent-plateau polynomials over k-Stirling permutations with initial letter 1 are γ-positive, a result that may have independent interest;
•
develop two crucial transformations on increasing pruned even k-ary forests to conclude our combinatorial interpretation.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.