Distribution of hooks in self-conjugate partitions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913786894483456 |
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| author | Craig, William Ono, Ken Singh, Ajit |
| author_facet | Craig, William Ono, Ken Singh, Ajit |
| contents | We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate partitions is asymptotically normally distributed with mean
$μ_t(n) \sim \frac{\sqrt{6n}}π + \frac{3}{π^2} - \frac{t}{2}+\frac{δ_t}{4}$ and variance $σ_t^2(n) \sim \frac{(π^2 - 6) \sqrt{6n}}{π^3},$ where $δ_t:=1$ if $t$ is odd, and is 0 otherwise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of hooks in self-conjugate partitions Craig, William Ono, Ken Singh, Ajit Combinatorics Number Theory We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate partitions is asymptotically normally distributed with mean $μ_t(n) \sim \frac{\sqrt{6n}}π + \frac{3}{π^2} - \frac{t}{2}+\frac{δ_t}{4}$ and variance $σ_t^2(n) \sim \frac{(π^2 - 6) \sqrt{6n}}{π^3},$ where $δ_t:=1$ if $t$ is odd, and is 0 otherwise. |
| title | Distribution of hooks in self-conjugate partitions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2406.09059 |