Rook decomposition of the Partition function

TL;DR

This paper introduces a new rook-based decomposition of the partition function, exploring connections with Durfee triangles, deriving generating functions, and analyzing properties like periodicity, parity, and growth asymptotics.

Contribution

It presents a novel rook decomposition of the partition function linked to Durfee triangles, including generating functions, recurrence relations, and asymptotic analysis.

Findings

Derived generating functions for specific Durfee triangle sizes

Established periodicity modulo p for partition counts

Analyzed parity and growth asymptotics of partitions

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.