Asymptotics of commuting ℓ -tuples in symmetric groups and log-concavity.

IF 0.6 Q3 MATHEMATICS Research in Number Theory Pub Date : 2024-01-01 Epub Date: 2024-10-03 DOI:10.1007/s40993-024-00562-1

Kathrin Bringmann, Johann Franke, Bernhard Heim

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Asymptotics of commuting <ns0:math><ns0:mi>ℓ</ns0:mi></ns0:math> -tuples in symmetric groups and log-concavity.","authors":"Kathrin Bringmann, Johann Franke, Bernhard Heim","doi":"10.1007/s40993-024-00562-1","DOIUrl":null,"url":null,"abstract":"Denote by <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> the number of <math><mi>ℓ</mi></math> -tuples of elements in the symmetric group <math><msub><mi>S</mi> <mi>n</mi></msub> </math> with commuting components, normalized by the order of <math><msub><mi>S</mi> <mi>n</mi></msub> </math> . In this paper, we prove asymptotic formulas for <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> . In addition, general criteria for log-concavity are shown, which can be applied to <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form <math><mrow><mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> <mi>c</mi> <mo>(</mo> <mi>b</mi> <mo>)</mo> <mo>></mo> <mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>)</mo></mrow> </math> for certain families of sequences c(n).","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11449981/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40993-024-00562-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/10/3 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Denote by $N_{ℓ} (n)$ the number of $ℓ$ -tuples of elements in the symmetric group $S_{n}$ with commuting components, normalized by the order of $S_{n}$ . In this paper, we prove asymptotic formulas for $N_{ℓ} (n)$ . In addition, general criteria for log-concavity are shown, which can be applied to $N_{ℓ} (n)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c (a) c (b) > c (a + b)$ for certain families of sequences c(n).

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

对称群中交换 ℓ -图元的渐近性和对数凹性

用 N ℓ ( n ) 表示对称群 S n 中具有交换成分的元素的 ℓ 元组数，以 S n 的阶归一化。本文证明了 N ℓ ( n ) 的渐近公式。此外，本文还展示了对数凹性的一般标准，这些标准可应用于 N ℓ ( n ) 及其他例子。此外，我们还得到了一个贝森罗特-奥诺（Bessenrodt-Ono）类型的定理，它给出了某些序列族 c(n) 的 c ( a ) c ( b ) > c ( a + b ) 的不等式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Research in Number Theory MATHEMATICS-

CiteScore

0.80

自引率

12.50%

发文量

期刊介绍： Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.