Small description about rook polynomial

1. ROOKS
POLYNOMIAL
BY : SAFEEQ K . K

3.  In chess a piece called a Rook or Castle is allowed at one turn to
be moved horizontally or vertically over as many unoccupied
spaces as one wishes
3 2 1
4
5 6
 Here a rook in square 3 of the figure can be move in one turn to
squares 1 , 2 or 4
 A rook at 5 can be moved to squares 2 or 6

4.  For k element of Z+ we want to determine the number of ways in
which k rooks can be placed on the unshaded squares of this
chessboard so that no two of them can be each other . That is
no two of them are in the same row or column in the board
 This number is denoted by rk or by rk (c)
3 2 1
4
5 6
 Two non - taking rooks can be placed at the following
pair of positions : {1,4} , {1,5} , {2,4} , {2,6} , {3,5} , {3,6} ,
{4,5} , {4,6} that is r2 = 8

5.  r3 = 2 using positions : {1,4,5} and {2,4,6}
The rooks polynomial is r0 + xr1 + x2r2 + · · ·
r(c,x) =1 + 6x + 8x2 + 2x3
In this case, one rook can be put any where, and
there are exactly two ways to place two rooks on the
board. The rooks polynomial is 1 + 4x + 2x2

6. The chessboard C is made up of 11 unshaded
squares.
C consist of 2 x 2 subboard C1 and a seven -
square subboard C2
These subboards are disjoint because they have
no squares in the same rows or column of C

7.  r(C1,x) = 1 + 4x + 2x2
 r(C2,x) = 1 + 7x + 10x2 + 2x3
 r(C,x) = 1 + 11x + 40x2 + 56x3 + 28x4 + 4x5
= r(C2,x) . r(C1,x)
In general disjoint subboards C1 , C2 ……..Cn
then r(C,x) = r(C1,x) r(C2,x) …… r(Cn,x)