Hook Length Biases in $t$-Core Partitions
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| Format: | Preprint |
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2026
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| author | Baruah, Nayandeep Deka Das, Hirakjyoti Mahanta, Pankaj Jyoti Saikia, Manjil P. |
| author_facet | Baruah, Nayandeep Deka Das, Hirakjyoti Mahanta, Pankaj Jyoti Saikia, Manjil P. |
| contents | Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in $\ell$-regular partitions. In this work, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n)\ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n)\ge a_{4,3}(n)$ for all $n$. The methods employed in this work are mainly combinatorial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10140 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hook Length Biases in $t$-Core Partitions Baruah, Nayandeep Deka Das, Hirakjyoti Mahanta, Pankaj Jyoti Saikia, Manjil P. Combinatorics Number Theory 05A17, 05A20, 11P81, 11P82 Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in $\ell$-regular partitions. In this work, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n)\ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n)\ge a_{4,3}(n)$ for all $n$. The methods employed in this work are mainly combinatorial. |
| title | Hook Length Biases in $t$-Core Partitions |
| topic | Combinatorics Number Theory 05A17, 05A20, 11P81, 11P82 |
| url | https://arxiv.org/abs/2603.10140 |