Alan Tucker's Applied Combinatorics question. I have a full $n \times n$ board and I need to find the rook polynomial. I am not sure if I comprehend the method wrong, but I am approaching this inductively, so I looked at a $2\times 2$ board (it's full so I'm assuming each square is shaded and I can place it in any way) so the polynomial I got for a $2\times 2$ board is $$ 1 + 4x + 4x^{2}.$$ Then for a $3\times 3$ board, I got $$ 1 + 9x + 12x^{2} + 22x^{3}.$$
However, I saw as comment on another question that my polynomial for 3x3 might be incorrect...
Anyway for a $4\times 4$ board, I'm at $$ 1 + 16x + 24x^{2} +...$$
Am I sort of on the right track as to how I would find the rook polynomial for a full nxn board?