A rook polynomial is a polynomial
 |
(1)
|
whose number of ways 
nonattacking rooks can be arranged on an 
chessboard. The rook
polynomials are given by
 |
(2)
|
where 
is an associated Laguerre polynomial.
The first few rook polynomials 
on square 
boards are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(OEIS A021010).
As an illustration, note that the 
case has two ways to place two rooks (i.e., the rook
number 
),
four ways to place one rook (
), and one way to place no rooks (
), hence 
.
