Rook Numbers, Partitions, and Durfee Triangles

We shall start the talk with an aptitude question: in how many ways can one place eight rooks on a chessboard such that no two rooks are attacking each other? Then, for a Ferrers board associated with a partition π, we ask its max-rook number R(π). This is shown to be the size of the Durfee triangle of π. We then discuss the generating function F_k(q) for the partitions of a positive integer n with a size-k Durfee triangle, denoted by R_k(n). We also obtain the asymptotic growth of R_k(n) as n→∞. We shall discuss a few other properties briefly. We conclude the talk by exploring possible future directions. This talk is based on my recent work with Armin Straub. This talk will be built on basic concepts of integer partitions and combinatorics.

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