The rook numbers are fairly well-studied in the literature. In this paper, we study the max-rook number of the Ferrers boards associated with 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 p ∈ N and p ≥ 2. 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.