Rook decomposition of the Partition function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912504491278336 |
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| author | Sharan, N. Guru |
| author_facet | Sharan, N. Guru |
| contents | The rook numbers are fairly well-studied in the literature. In this paper, we study the max-rook number of the Ferrers boards associated to integer partitions. We show its connections with the Durfee triangle of the partitions. The max-rook number gives a new decomposition of the partition function. We derive the generating functions of the partitions with the Durfee triangle of sizes $3$, $4$ and $5$. We obtain their exact formula and further use it to show the periodicity modulo $p$ for any $p \in \mathbb{N}$ and $p\geq2$. We also establish their parity and parity bias. We give the growth asymptotics of partitions with the Durfee triangle of sizes $3$ and $4$. We obtain a new rook analogue of the recurrence relation of the partition function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rook decomposition of the Partition function Sharan, N. Guru Combinatorics Number Theory Primary 05A17, Secondary 11P83, 11N56 The rook numbers are fairly well-studied in the literature. In this paper, we study the max-rook number of the Ferrers boards associated to integer partitions. We show its connections with the Durfee triangle of the partitions. The max-rook number gives a new decomposition of the partition function. We derive the generating functions of the partitions with the Durfee triangle of sizes $3$, $4$ and $5$. We obtain their exact formula and further use it to show the periodicity modulo $p$ for any $p \in \mathbb{N}$ and $p\geq2$. We also establish their parity and parity bias. We give the growth asymptotics of partitions with the Durfee triangle of sizes $3$ and $4$. We obtain a new rook analogue of the recurrence relation of the partition function. |
| title | Rook decomposition of the Partition function |
| topic | Combinatorics Number Theory Primary 05A17, Secondary 11P83, 11N56 |
| url | https://arxiv.org/abs/2507.20260 |