Rooks 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)