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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2403.05072 |
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| _version_ | 1866915242185850880 |
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| author | Menon, Krishna |
| author_facet | Menon, Krishna |
| contents | It is known that for the Young diagram of any partition of an integer $n$, the sum of squares of the hook lengths of its cells is exactly $n^2$ more than that of the contents of its cells. That is, for any partition $λ$ of an integer $n$, \begin{equation*}
\sum_{u \in λ} h(u)^2 = n^2 + \sum_{u \in λ} c(u)^2. \end{equation*} We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sum of squares of hook lengths and contents Menon, Krishna Combinatorics 05A17, 05A19 It is known that for the Young diagram of any partition of an integer $n$, the sum of squares of the hook lengths of its cells is exactly $n^2$ more than that of the contents of its cells. That is, for any partition $λ$ of an integer $n$, \begin{equation*} \sum_{u \in λ} h(u)^2 = n^2 + \sum_{u \in λ} c(u)^2. \end{equation*} We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents. |
| title | Sum of squares of hook lengths and contents |
| topic | Combinatorics 05A17, 05A19 |
| url | https://arxiv.org/abs/2403.05072 |