ABSTRACT
We have seen how to enumerate circular permutations in Chapter 2; the idea there is that two configurations are considered identical if one may be transformed into another by means of a rotation. There are other possible ways for two configurations to be equivalent, and in this chapter we look at the ways to enumerate configurations that are distinct with respect to some collection of symmetries. We will refer to these configurations as “colorings,” even though what is meant by “color” might be nothing that an artist or decorator would recognize. For instance, we may seat politicians at a circular table, and only care which of (say) three political leanings (conservative, moderate, or liberal) the politician has. Thus, we might label seats with the “colors” C, M, and L. If we are concerned with gender, we might use “colors” M and F; and with religion or ethnicity, there would be many possible labels. For class-scheduling purposes, a classroom might get a label indicating that it can accommodate large classes or that it is equipped with a computer projector.
