The Pólya enumeration theorem is a combinatorial theorem that provides a way to count the distinct arrangements (or colorings) of objects under group actions, particularly useful in situations where symmetries play a role. Named after mathematician George Pólya, the theorem is a powerful tool in combinatorial enumeration, especially in counting labeled and unlabeled structures that exhibit symmetry.
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The Pólya enumeration theorem counts colourings up to a finite group action using the action's cycle index. Substituting colour weights into each cycle variable groups colourings by their colour inventories.