The rook numbers r_k^(m, n) of an m×n board are the number of subsets of size k such that no two elements have the same first or second coordinate. In other word, it is the number of ways of placing k rooks on a board such that none attack each other (one form of the so-called rooks problem). The rook number r_k is therefore the leading coefficient of the corresponding rook polynomial R_(m n)(x).

permutation matrix | rook polynomial | rooks problem