{"title":"渐近性,Turán不等式,以及分区的bg秩和2商秩的分布","authors":"Andrew Baker, Joshua Males","doi":"10.1007/s00026-022-00612-4","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>j</i>, <i>n</i> be even positive integers, and let <span>\\(\\overline{p}_j(n)\\)</span> denote the number of partitions with BG-rank <i>j</i>, and <span>\\(\\overline{p}_j(a,b;n)\\)</span> to be the number of partitions with BG-rank <i>j</i> and 2-quotient rank congruent to <span>\\(a \\ \\, \\left( \\mathrm {mod} \\, b \\right) \\)</span>. We give asymptotics for both statistics, and show that <span>\\(\\overline{p}_j(a,b;n)\\)</span> is asymptotically equidistributed over the congruence classes modulo <i>b</i>. We also show that each of <span>\\(\\overline{p}_j(n)\\)</span> and <span>\\(\\overline{p}_j(a,b;n)\\)</span> asymptotically satisfy all higher-order Turán inequalities.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions\",\"authors\":\"Andrew Baker, Joshua Males\",\"doi\":\"10.1007/s00026-022-00612-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>j</i>, <i>n</i> be even positive integers, and let <span>\\\\(\\\\overline{p}_j(n)\\\\)</span> denote the number of partitions with BG-rank <i>j</i>, and <span>\\\\(\\\\overline{p}_j(a,b;n)\\\\)</span> to be the number of partitions with BG-rank <i>j</i> and 2-quotient rank congruent to <span>\\\\(a \\\\ \\\\, \\\\left( \\\\mathrm {mod} \\\\, b \\\\right) \\\\)</span>. We give asymptotics for both statistics, and show that <span>\\\\(\\\\overline{p}_j(a,b;n)\\\\)</span> is asymptotically equidistributed over the congruence classes modulo <i>b</i>. We also show that each of <span>\\\\(\\\\overline{p}_j(n)\\\\)</span> and <span>\\\\(\\\\overline{p}_j(a,b;n)\\\\)</span> asymptotically satisfy all higher-order Turán inequalities.</p></div>\",\"PeriodicalId\":50769,\"journal\":{\"name\":\"Annals of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00026-022-00612-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-022-00612-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions
Let j, n be even positive integers, and let \(\overline{p}_j(n)\) denote the number of partitions with BG-rank j, and \(\overline{p}_j(a,b;n)\) to be the number of partitions with BG-rank j and 2-quotient rank congruent to \(a \ \, \left( \mathrm {mod} \, b \right) \). We give asymptotics for both statistics, and show that \(\overline{p}_j(a,b;n)\) is asymptotically equidistributed over the congruence classes modulo b. We also show that each of \(\overline{p}_j(n)\) and \(\overline{p}_j(a,b;n)\) asymptotically satisfy all higher-order Turán inequalities.
期刊介绍:
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches