Rook Theory. II: Boards of Binomial Type

Abstract

We explore the relation between rook theory of Ferrers boards and polynomial sequences of binomial type. Recursion and explicit formulas for the rook numbers of Riordan’s trapezoidal boards are derived. It is shown that some classic sequences of rook and factorial polynomials are of binomial type. A class of boards corresponding to Abel’s theorem and its generalizations are constructed. The results are developed from both algebraic and combinatorial viewpoints.