On hook length biases in $t$-regular partitions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910776776720384 |
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| author | Barman, Rupam Mahanta, Pankaj Jyoti Singh, Gurinder |
| author_facet | Barman, Rupam Mahanta, Pankaj Jyoti Singh, Gurinder |
| contents | Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On hook length biases in $t$-regular partitions Barman, Rupam Mahanta, Pankaj Jyoti Singh, Gurinder Combinatorics Number Theory 05A17 (Primary), 05A15, 11P81, 11P82 (Secondary) Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$. |
| title | On hook length biases in $t$-regular partitions |
| topic | Combinatorics Number Theory 05A17 (Primary), 05A15, 11P81, 11P82 (Secondary) |
| url | https://arxiv.org/abs/2412.00973 |