A note on Diagonal sequences of integer partitions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929621088337920 |
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| author | Neubauer, Michael Vargas, Harmony |
| author_facet | Neubauer, Michael Vargas, Harmony |
| contents | Let \(\mathcal{P}(n)\) be the set of partitions of the positive integer \(n\). For \(α=(α_1,...,α_t) \in \mathcal{P}(n)\) define the diagonal sequence \(δ(α)=(d_k(α))_{k \geq 1}\) via \( d_k(α) = \big\lvert \{ i \, \rvert \, 1 \leq i \leq k \mbox{ and } α_i + i- 1\geq k \} \big\rvert.\) We show that the set of all partitions in \(\mathcal{P}(n)\) with the same diagonal sequence is a partially ordered set under majorization with unique maximal and minimal elements and we give an explicit formula for the number of partitions with the same diagonal sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on Diagonal sequences of integer partitions Neubauer, Michael Vargas, Harmony Combinatorics Let \(\mathcal{P}(n)\) be the set of partitions of the positive integer \(n\). For \(α=(α_1,...,α_t) \in \mathcal{P}(n)\) define the diagonal sequence \(δ(α)=(d_k(α))_{k \geq 1}\) via \( d_k(α) = \big\lvert \{ i \, \rvert \, 1 \leq i \leq k \mbox{ and } α_i + i- 1\geq k \} \big\rvert.\) We show that the set of all partitions in \(\mathcal{P}(n)\) with the same diagonal sequence is a partially ordered set under majorization with unique maximal and minimal elements and we give an explicit formula for the number of partitions with the same diagonal sequence. |
| title | A note on Diagonal sequences of integer partitions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.06856 |