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arXiv · 2507.20260

Rook decomposition of the Partition function

N. Guru Sharan
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Abstract

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.

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BibTeXRIS

N. Guru Sharan. 2025-07-27. Rook decomposition of the Partition function. https://arxiv.org/abs/2507.20260

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