{"title":"超平面上 Sn 轨道的分数","authors":"Brendan Pawlowski","doi":"10.1016/j.laa.2024.07.022","DOIUrl":null,"url":null,"abstract":"<div><p>Huang, McKinnon, and Satriano conjectured that if <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> has distinct coordinates and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, then a hyperplane through the origin other than <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> contains at most <span><math><mn>2</mn><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>!</mo></math></span> of the vectors obtained by permuting the coordinates of <em>v</em>. We prove this conjecture.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The fraction of an Sn-orbit on a hyperplane\",\"authors\":\"Brendan Pawlowski\",\"doi\":\"10.1016/j.laa.2024.07.022\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Huang, McKinnon, and Satriano conjectured that if <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> has distinct coordinates and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, then a hyperplane through the origin other than <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> contains at most <span><math><mn>2</mn><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>!</mo></math></span> of the vectors obtained by permuting the coordinates of <em>v</em>. We prove this conjecture.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003124\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003124","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Huang, McKinnon, and Satriano conjectured that if has distinct coordinates and , then a hyperplane through the origin other than contains at most of the vectors obtained by permuting the coordinates of v. We prove this conjecture.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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