Let $G = (V,E)$ be a graph. Denote by $N_k$ the number of partitions of $V$ that give $k$ independent sets. Then we have that the chromatic polynomial is $\pi_G(x) = \sum_{k = 1}^n N_k x(x-1)(x-2)\ldots (x-k+1).$
For the path with $4$ vertices, $P_4$, we have that $N_1 = 0, N_2 = 1, N_3 = 2, N_4 = 1$. Then by the sum above $\pi_{P_4}(x) = x^4 - 4 x^3 + 6 x^2 - 3 x$. But the chromatic polynomial is supposed to be $x(x-1)^3$. What's wrong?