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

For an board, each permutation matrix corresponds to an allowed configuration of rooks. However, the permutation matrices give only a subset of the total number of solutions, which on an board is simply the factorial . This can be seen easily by noting that there are ways to place the first rook in the first column, ways to place the second rook in the second column, ways to place the third rook, ..., and a single way to place the th rook in the last (th) column.

The rook numbers of a board determine the rook numbers of the complementary board , written as . This is known as the rook reciprocity theorem.