Abstract
This chapter explores Polya's pattern analysis. Polya's Pattern Inventory [Polya] is concerned with the following combinatorial problem. Suppose there is a pattern consisting of n regions that are to be colored using m colors. Polya's Pattern Inventory answers the general question: if there is a group of symmetries acting on the n regions, are two colorings by m colors to be considered equivalent if one coloring is taken to the other by one of the symmetries? This question can be investigated from a geometrical or an algebraic point of view. The algebraic approach, discovered by Polya, consists of the construction of a polynomial from which one can read off how many colorings there are for specific numbers of regions of specified colors. It concentrates solely on finding these numbers and never does spell out what the actual patterns are. These numbers will appear as coefficients in a polynomial whose variables represent the colors.
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