Pólya enumeration theorem

The Pólya enumeration theorem is a mathematical tool used to count the number of distinct arrangements or configurations of objects when symmetry (like rotations or reflections) makes some arrangements equivalent. For example, it helps count unique colorings of a shape where rotating it doesn't produce a new, different coloring. It combines group theory and combinatorics to efficiently account for these symmetries, providing a systematic way to find total distinguishable patterns without counting identical ones repeatedly. This theorem is useful in chemistry, physics, and computer science for analyzing symmetrical structures and patterns.