Invisible permutations and rook placements on a Ferrers board
We study non-attacking rook placements on a Ferrers board by extending them to certain permutation matrices, called invisible permutations. By introducing a length function on rook placements, we define a rook length polynomial \(\text{RL}_ k(\lambda, q)\) for a given number \(k\) of rooks and a Ferrers board \(F_ \lambda\). We prove that \(\text{R}_ k(\lambda, q)= q^{C_{\lambda,k}} \text{RL}_ k(\lambda, q)\), where \(\text{R}_ k(\lambda, q)\) is a Garsia-Remmel polynomial. Further, we give explicit formulae for Garsia-Remmel polynomials and rook length polynomials.
scientific article; zbMATH DE number 773905
scientific article