In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged. The term "rook polynomial" was coined by John Riordan. Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board B that is a subset of the n × n chessboard corresponds to permutations of n objects, which we may take to be the numbers 1, 2, ..., n, such that the number aj in the j-th position in the permutation must be the column number of an allowed square in row j of B. Famous examples include the number of ways to place n non-attacking rooks on:
an entire n × n chessboard, which is an elementary combinatorial problem; the same board with its diagonal squares forbidden; this is the derangement or "hat-check" problem (this is a particular case of the problème des rencontres); the same board without the squares on its diagonal and immediately above its diagonal (and without the bottom left square), which is essential in the solution of the problème des ménages. Interest in rook placements arises in pure and applied combinatorics, group theory, number theory, and statistical physics. The particular value of rook polynomials comes from the utility of the generating function approach, and also from the fact that the zeroes of the rook polynomial of a board provide valuable information about its coefficients, i.e., the number of non-attacking placements of k rooks.
Definition The rook polynomial RB(x) of a board B is the generating function for the numbers of arrangements of non-attacking rooks:
R B ( x ) = ∑ k = 0 min ( m , n ) r k ( B ) x k , {\displaystyle R_{B}(x)=\sum _{k=0}^{\min {(m,n)}}r_{k}(B)x^{k},}
where r k ( B ) {\displaystyle r_{k}(B)} is the number of ways to place k non-attacking rooks on the board B. There is a maximum number of non-attacking rooks the board can hold; indeed, there cannot be more rooks than the number of rows or number of columns in the board (hence the limit min ( m , n ) {\displaystyle \min(m,n)} ).
Complete boards For rectangular m × n boards Bm,n, we write Rm,n := RBm,n, and if m=n, Rn := Rm,n. The first few rook polynomials on square n × n boards are:
R 1 ( x ) = x + 1 R 2 ( x ) = 2 x 2 + 4 x + 1 R 3 ( x ) = 6 x 3 + 18 x 2 + 9 x + 1 R 4 ( x ) = 24 x 4 + 96 x 3 + 72 x 2 + 16 x + 1. {\displaystyle {\begin{aligned}R_{1}(x)&=x+1\\R_{2}(x)&=2x^{2}+4x+1\\R_{3}(x)&=6x^{3}+18x^{2}+9x+1\\R_{4}(x)&=24x^{4}+96x^{3}+72x^{2}+16x+1.\end{aligned}}}
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