A rook polynomial is a polynomial whose coefficient of 
counts the ways to place 
nonattacking rooks on a specified board. For a complete 
chessboard,
where 
and 
are nonnegative integers, it is
 |
(1)
|
where 
.
These rook polynomials are given by
 |
(2)
|
where 
is an associated Laguerre polynomial
and the expression at 
is understood by polynomial continuation.
The first few rook polynomials 
on square 
boards are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
whose coefficients are the absolute values of the corresponding rows of 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 
.
