On the distribution of $t$-hooks of doubled distinct partitions
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910877112860672 |
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| author | Cho, Hyunsoo Kim, Byungchan Kim, Eunmi Yee, Ae Ja |
| author_facet | Cho, Hyunsoo Kim, Byungchan Kim, Eunmi Yee, Ae Ja |
| contents | Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate partitions of $n$. Motivated by these studies, in this paper, we further investigate the number of $t$-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions are both asymptotically normally distributed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the distribution of $t$-hooks of doubled distinct partitions Cho, Hyunsoo Kim, Byungchan Kim, Eunmi Yee, Ae Ja Combinatorics Number Theory 05A17, 11P82 Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate partitions of $n$. Motivated by these studies, in this paper, we further investigate the number of $t$-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of $t$-hooks in doubled distinct partitions and the number of $t$-shifted hooks in strict partitions are both asymptotically normally distributed. |
| title | On the distribution of $t$-hooks of doubled distinct partitions |
| topic | Combinatorics Number Theory 05A17, 11P82 |
| url | https://arxiv.org/abs/2503.12040 |