How many paths can a Rook travels from upper Left Corner to lower right corner : Well you can find a solution at http://mathacadabra.com/Items2013/GeneratingFunctionAdventuresIV.aspx it needs a little bit some explanations first about the case 8 * 8 :

- First the rook will start at a8, then to reach a8, there is only one path : staying at the same place
- the chess rook can move horizontally or vertically from 1 up to 8 (including is initial position)
- the table sum up the number of paths could be used by the rook in order to reach the case, for example if the rook want to reach e5 ( 838) , the rook can come only vertically from the case above e4 ( e5, e6, e7, e8) or horizontally from the left of e4 ( a4, b4, c4, d4 ) therefore to get the total number of paths to reach e4 is : 8+28+94+289+289+94+28 = 838 paths for a square 5 * 5, we can calculate all the paths recursively
- Actually there are 470010 paths for a square 8*8
- In the Link provided above to the website , they are trying to calculate the generating function for the diagonal which is the number of paths for a square N * N.
- let’s demonstrate after reading the post :( http://mathacadabra.com/Items2013/GeneratingFunctionAdventuresIV.aspx ) that the 2 variables generating function is (1 – s – t + st) / ( 1 – 2 *s -2*t + 3 *st ) for a square N*N