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We have talked about the number of ways to properly color a graph with $$k$$ colors, given by the chromatic polynomial. For example, the chromatic polynomial for the graph in Figure 6.0.1 is $$k^4 - 4k^3 + 6k^2 - 3k$$, and $$f(2)=2$$. The two colorings are shown in the figure, but in an obvious sense they are the same coloring, since one can be turned into the other by simply rotating the graph. We will consider a slightly different sort of coloring problem, in which we count the "truly different'' colorings of objects.
Figure 6.0.1. $$C_4$$ drawn as a square, colored in two ways.