Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy
{"title":"将整数分割为小幂的中心极限定理","authors":"Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy","doi":"10.1007/s00605-023-01926-y","DOIUrl":null,"url":null,"abstract":"<p>The study of the well-known partition function <i>p</i>(<i>n</i>) counting the number of solutions to <span>\\(n = a_{1} + \\dots + a_{\\ell }\\)</span> with integers <span>\\(1 \\le a_{1} \\le \\dots \\le a_{\\ell }\\)</span> has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into </p><span>$$\\begin{aligned} n=\\left\\lfloor a_1^\\alpha \\right\\rfloor +\\cdots +\\left\\lfloor a_\\ell ^\\alpha \\right\\rfloor \\end{aligned}$$</span><p>with <span>\\(1\\le a_1< \\cdots < a_\\ell \\)</span> and some fixed <span>\\(0< \\alpha < 1\\)</span>. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A central limit theorem for integer partitions into small powers\",\"authors\":\"Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy\",\"doi\":\"10.1007/s00605-023-01926-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The study of the well-known partition function <i>p</i>(<i>n</i>) counting the number of solutions to <span>\\\\(n = a_{1} + \\\\dots + a_{\\\\ell }\\\\)</span> with integers <span>\\\\(1 \\\\le a_{1} \\\\le \\\\dots \\\\le a_{\\\\ell }\\\\)</span> has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into </p><span>$$\\\\begin{aligned} n=\\\\left\\\\lfloor a_1^\\\\alpha \\\\right\\\\rfloor +\\\\cdots +\\\\left\\\\lfloor a_\\\\ell ^\\\\alpha \\\\right\\\\rfloor \\\\end{aligned}$$</span><p>with <span>\\\\(1\\\\le a_1< \\\\cdots < a_\\\\ell \\\\)</span> and some fixed <span>\\\\(0< \\\\alpha < 1\\\\)</span>. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-023-01926-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01926-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A central limit theorem for integer partitions into small powers
The study of the well-known partition function p(n) counting the number of solutions to \(n = a_{1} + \dots + a_{\ell }\) with integers \(1 \le a_{1} \le \dots \le a_{\ell }\) has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into
with \(1\le a_1< \cdots < a_\ell \) and some fixed \(0< \alpha < 1\). In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.