Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910910243667968 |
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| author | Kim, Eunmi |
| author_facet | Kim, Eunmi |
| contents | In this article, we study hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions. More precisely, we establish hook length inequalities between $\ell$-regular partitions and $\ell$-distinct partitions for hook lengths $2$ and $3$, by deriving asymptotic formulas for the total number of hooks of length $t$ in both partition classes, for $t = 1, 2, 3$. From these asymptotics, we show that the ratio of the total number of hooks of length $t$ in $\ell$-regular partitions to those in $\ell$-distinct partitions tends to a constant that depends on $\ell$ and $t$. We also provide hook length inequalities within $\ell$-regular partitions and within $\ell$-distinct partitions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions Kim, Eunmi Combinatorics Number Theory 05A17, 11P82 In this article, we study hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions. More precisely, we establish hook length inequalities between $\ell$-regular partitions and $\ell$-distinct partitions for hook lengths $2$ and $3$, by deriving asymptotic formulas for the total number of hooks of length $t$ in both partition classes, for $t = 1, 2, 3$. From these asymptotics, we show that the ratio of the total number of hooks of length $t$ in $\ell$-regular partitions to those in $\ell$-distinct partitions tends to a constant that depends on $\ell$ and $t$. We also provide hook length inequalities within $\ell$-regular partitions and within $\ell$-distinct partitions. |
| title | Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions |
| topic | Combinatorics Number Theory 05A17, 11P82 |
| url | https://arxiv.org/abs/2501.10916 |