Pólya’s enumeration theorem

Pólya’s enumeration theorem is a mathematical method used to count distinct arrangements—like colorings or patterns—of objects when some symmetries (like rotations or reflections) make certain arrangements equivalent. Instead of counting each pattern individually, it considers how the symmetries group similar arrangements together, streamlining the counting process. This approach is useful in chemistry, combinatorics, and design, helping to determine the number of unique configurations without exhaustive listing. Essentially, it simplifies the problem of counting objects that look different but are considered the same due to their symmetrical properties.