Cores of partitions in rectangles
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916225184956416 |
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| author | Ayyer, Arvind Sinha, Shubham |
| author_facet | Ayyer, Arvind Sinha, Shubham |
| contents | For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $r \times s$ rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $ρ$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $α= (t-1)/2$ and rate parameter $β$ that depends on the aspect ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07996 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cores of partitions in rectangles Ayyer, Arvind Sinha, Shubham Combinatorics Probability 05A15, 05A16, 05A17, 60C05 For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $r \times s$ rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $ρ$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $α= (t-1)/2$ and rate parameter $β$ that depends on the aspect ratio. |
| title | Cores of partitions in rectangles |
| topic | Combinatorics Probability 05A15, 05A16, 05A17, 60C05 |
| url | https://arxiv.org/abs/2211.07996 |