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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.16185 |
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| _version_ | 1866916373793341440 |
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| author | Liu, Xin-Bei Zhang, Zhong-Xue |
| author_facet | Liu, Xin-Bei Zhang, Zhong-Xue |
| contents | A sequence of $\{a_n\}_{n\ge 0}$ satisfies the Briggs inequality if \begin{align*} a_n^2(a_n^2-a_{n-1}a_{n+1})>a_{n-1}^2(a_{n+1}^2-a_na_{n+2}) \end{align*} holds for any $n\ge 1$. In this paper we show that both the partition function $\{p(n+N_0)\}_{n\geq 0}$ and the overpartition function $\{\overline{p}(n+\overline{N}_0)\}_{n\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\overline{N}_{0}$. Based on Chern's formula for $η$-quotients, we further prove that the $k$-regular partition function $\{p_k(n+N_{k})\}_{n\geq 0}$ and the $k$-regular overpartition function $\{\overline{p}_k(n+\overline{N}_k)\}_{n\ge 0}$ also satisfy the Briggs inequality for $2\le k\le 9$ and some $N_k,\overline{N}_{k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Briggs inequality for partitions and overpartitions Liu, Xin-Bei Zhang, Zhong-Xue Combinatorics A sequence of $\{a_n\}_{n\ge 0}$ satisfies the Briggs inequality if \begin{align*} a_n^2(a_n^2-a_{n-1}a_{n+1})>a_{n-1}^2(a_{n+1}^2-a_na_{n+2}) \end{align*} holds for any $n\ge 1$. In this paper we show that both the partition function $\{p(n+N_0)\}_{n\geq 0}$ and the overpartition function $\{\overline{p}(n+\overline{N}_0)\}_{n\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\overline{N}_{0}$. Based on Chern's formula for $η$-quotients, we further prove that the $k$-regular partition function $\{p_k(n+N_{k})\}_{n\geq 0}$ and the $k$-regular overpartition function $\{\overline{p}_k(n+\overline{N}_k)\}_{n\ge 0}$ also satisfy the Briggs inequality for $2\le k\le 9$ and some $N_k,\overline{N}_{k}$. |
| title | The Briggs inequality for partitions and overpartitions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.16185 |