Question 144
Two colorings are distinct if there is no way to relabel the colors to transform one into the other. Equivalently, they are distinct if and only if there is some pair of vertices which are the same color in one coloring but different colors in the other. For what pairs $(n, k)$ of positive integers does there exist a finite graph with chromatic number $n$ which has exactly $k$ distinct good colorings using $n$ colors?