{"title":"过分割秩的严格对数子可加性","authors":"Helen W. J. Zhang, Ying Zhong","doi":"10.1007/s00026-023-00643-5","DOIUrl":null,"url":null,"abstract":"<div><p>Bessenrodt and Ono initially found the strict log-subadditivity of partition function <i>p</i>(<i>n</i>), that is, <span>\\(p(a+b)< p(a)p(b)\\)</span> for <span>\\(a,b>1\\)</span> and <span>\\(a+b>9\\)</span>. Many other important partition statistics are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson’s rank for partitions from the <i>q</i>-series perspective. Let <span>\\({\\overline{N}}(a,c,n)\\)</span> denote the number of overpartitions with rank congruent to <i>a</i> modulo <i>c</i>. Ciolan computed the asymptotic formula of <span>\\({\\overline{N}}(a,c,n)\\)</span> and showed that <span>\\({\\overline{N}}(a, c, n) > {\\overline{N}}(b, c, n)\\)</span> for <span>\\(0\\le a<b\\le \\lfloor \\frac{c}{2}\\rfloor \\)</span> and <i>n</i> large enough if <span>\\(c\\ge 7\\)</span>. In this paper, we derive an upper bound and a lower bound of <span>\\({\\overline{N}}(a,c,n)\\)</span> for each <span>\\(c\\ge 3\\)</span> by using the asymptotics due to Ciolan. Consequently, we establish the strict log-subadditivity of <span>\\({\\overline{N}}(a,c,n)\\)</span> analogous to the partition function <i>p</i>(<i>n</i>).</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"27 4","pages":"799 - 817"},"PeriodicalIF":0.6000,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Strict Log-Subadditivity for Overpartition Rank\",\"authors\":\"Helen W. J. Zhang, Ying Zhong\",\"doi\":\"10.1007/s00026-023-00643-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Bessenrodt and Ono initially found the strict log-subadditivity of partition function <i>p</i>(<i>n</i>), that is, <span>\\\\(p(a+b)< p(a)p(b)\\\\)</span> for <span>\\\\(a,b>1\\\\)</span> and <span>\\\\(a+b>9\\\\)</span>. Many other important partition statistics are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson’s rank for partitions from the <i>q</i>-series perspective. Let <span>\\\\({\\\\overline{N}}(a,c,n)\\\\)</span> denote the number of overpartitions with rank congruent to <i>a</i> modulo <i>c</i>. Ciolan computed the asymptotic formula of <span>\\\\({\\\\overline{N}}(a,c,n)\\\\)</span> and showed that <span>\\\\({\\\\overline{N}}(a, c, n) > {\\\\overline{N}}(b, c, n)\\\\)</span> for <span>\\\\(0\\\\le a<b\\\\le \\\\lfloor \\\\frac{c}{2}\\\\rfloor \\\\)</span> and <i>n</i> large enough if <span>\\\\(c\\\\ge 7\\\\)</span>. In this paper, we derive an upper bound and a lower bound of <span>\\\\({\\\\overline{N}}(a,c,n)\\\\)</span> for each <span>\\\\(c\\\\ge 3\\\\)</span> by using the asymptotics due to Ciolan. Consequently, we establish the strict log-subadditivity of <span>\\\\({\\\\overline{N}}(a,c,n)\\\\)</span> analogous to the partition function <i>p</i>(<i>n</i>).</p></div>\",\"PeriodicalId\":50769,\"journal\":{\"name\":\"Annals of Combinatorics\",\"volume\":\"27 4\",\"pages\":\"799 - 817\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-03-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00026-023-00643-5\",\"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/s00026-023-00643-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Bessenrodt and Ono initially found the strict log-subadditivity of partition function p(n), that is, \(p(a+b)< p(a)p(b)\) for \(a,b>1\) and \(a+b>9\). Many other important partition statistics are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson’s rank for partitions from the q-series perspective. Let \({\overline{N}}(a,c,n)\) denote the number of overpartitions with rank congruent to a modulo c. Ciolan computed the asymptotic formula of \({\overline{N}}(a,c,n)\) and showed that \({\overline{N}}(a, c, n) > {\overline{N}}(b, c, n)\) for \(0\le a<b\le \lfloor \frac{c}{2}\rfloor \) and n large enough if \(c\ge 7\). In this paper, we derive an upper bound and a lower bound of \({\overline{N}}(a,c,n)\) for each \(c\ge 3\) by using the asymptotics due to Ciolan. Consequently, we establish the strict log-subadditivity of \({\overline{N}}(a,c,n)\) analogous to the partition function p(n).
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