Dual rook polynomials Original Research Article

A theorem contained in the paper ‘A combinatoric formula’ by Wang, Lee and Tan (J. Math. Anal. Appl. 160 (1991) 500–503) gives rise to the definition of certain polynomials associated with boards: Stimulated by the analogy to the well-known rook polynomials, we call them ‘dual rook polynomials’. We show that in the cases of Ferrers boards and skew boards the evaluation of these polynomials at −1 always yields values −1, 0 or 1, generalizing the theorem cited above. Moreover, we evaluate these polynomials at −2. Finally, we state three conjectures that are quite well supported by empirical tests: Two of these conjectures are known to be true for rook polynomials.