A Central Limit Theorem for Integer Partitions into Small Powers
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917558124281856 |
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| author | Lipnik, Gabriel F. Madritsch, Manfred G. Tichy, Robert F. |
| author_facet | Lipnik, Gabriel F. Madritsch, Manfred G. Tichy, Robert F. |
| contents | The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a variant, namely partitions of integers into \begin{equation*}
n=\lfloor a_1^α\rfloor + \cdots + \lfloor a_\ell^α\rfloor \end{equation*} with $1\leq a_1 < \cdots < a_\ell$ and some fixed $0 < α< 1$. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle point method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05592 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Central Limit Theorem for Integer Partitions into Small Powers Lipnik, Gabriel F. Madritsch, Manfred G. Tichy, Robert F. Number Theory Combinatorics 11P82, 05A17, 60F05 The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a variant, namely partitions of integers into \begin{equation*} n=\lfloor a_1^α\rfloor + \cdots + \lfloor a_\ell^α\rfloor \end{equation*} with $1\leq a_1 < \cdots < a_\ell$ and some fixed $0 < α< 1$. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle point method. |
| title | A Central Limit Theorem for Integer Partitions into Small Powers |
| topic | Number Theory Combinatorics 11P82, 05A17, 60F05 |
| url | https://arxiv.org/abs/2204.05592 |