Log-concavity And The Multiplicative Properties of Restricted Partition Functions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909606577438720 |
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| author | Roy, Arindam |
| author_facet | Roy, Arindam |
| contents | The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq p(n+m)$ for $n\geq m\geq 2$ with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $\{x_n\}$ satisfying a particular initial condition likewise satisfies the inequality $x_nx_m \geq x_{n+m}$. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log-concavity And The Multiplicative Properties of Restricted Partition Functions Roy, Arindam Number Theory The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq p(n+m)$ for $n\geq m\geq 2$ with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $\{x_n\}$ satisfying a particular initial condition likewise satisfies the inequality $x_nx_m \geq x_{n+m}$. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation. |
| title | Log-concavity And The Multiplicative Properties of Restricted Partition Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.03153 |