Polya Enumeration Calculator

About this calculator

Pólya enumeration counts how many genuinely distinct ways there are to color n positions arranged in a circle with k colors, treating rotations of the same pattern as identical rather than counting them separately. It works via Burnside's lemma: the count equals the average, over every rotation, of the number of colorings that rotation leaves unchanged — which simplifies to (1/n)·Σ φ(d)·k^(n/d) summed over the divisors d of n, using Euler's totient function φ.

This is the classic “how many distinct necklaces” problem from combinatorics, and the same math shows up wherever you need to count arrangements under a symmetry group — computer scientists counting distinct circular data structures, and chemists counting distinct molecular isomers of ring-shaped compounds where rotating the ring doesn't create a new molecule.