Rook decomposition of the partition function

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 , and . We obtain their exact formula and further use it to show the periodicity modulo for any and . We also establish their parity and parity bias. We give the growth asymptotics of partitions with the Durfee triangle of sizes and . We obtain a new rook analogue of the recurrence relation of the partition function.