The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912355109044224 |
|---|---|
| author | Lin, Hongshu Zang, Wenston J. T. |
| author_facet | Lin, Hongshu Zang, Wenston J. T. |
| contents | Let $b_{n,k}$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh and Barman raised the question of finding the relation between $b_{t,2}(n)$ and $b_{t,1}(n)$. Kim showed that there exists $N$ such that $b_{t,2}(n)\ge b_{t,1}(n)$ and $b_{t,2}(n) \geq b_{t,3}(n)$ for $n>N$. In this paper, we find an explicit bound of $N=O(t^5)$ for $b_{t,2}(n)\geq b_{t,1}(n)$ and show that $b_{t,2}(n) \geq b_{t,3}(n)$ for all $n\ge 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions Lin, Hongshu Zang, Wenston J. T. Combinatorics Number Theory Let $b_{n,k}$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh and Barman raised the question of finding the relation between $b_{t,2}(n)$ and $b_{t,1}(n)$. Kim showed that there exists $N$ such that $b_{t,2}(n)\ge b_{t,1}(n)$ and $b_{t,2}(n) \geq b_{t,3}(n)$ for $n>N$. In this paper, we find an explicit bound of $N=O(t^5)$ for $b_{t,2}(n)\geq b_{t,1}(n)$ and show that $b_{t,2}(n) \geq b_{t,3}(n)$ for all $n\ge 4$. |
| title | The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2504.21655 |