Polyacounters

Polyacounters is an iOS tool designed for learning combinatorics and counting theory, helping you understand the classic problem of "how many fundamentally different coloring schemes exist" under the influence of symmetry groups. In Workbench, you can set the number of vertices n and available colors k for a regular n-sided shape, and select the type of group action - identity group, cyclic group C ₙ, or dihedral group D. The application automatically calculates the number of orbits (equivalent classes) based on Burnside lemma, displays the fixed point contribution of each group element, and provides the complete counting result through P ó lya enumeration theorem and cycle index. When the parameters are appropriate, the application will enumerate all equivalent classes and display the representative coloring of each class in a colored polygon graph, intuitively demonstrating how symmetry is merged. The Elements page supports interactive exploration: select group element g, edit vertex shading and apply permutation with one click, observe how the shape changes under symmetry, and view cycle notation and | Fix (g) | details. The Compare page allows for parallel comparison of the orbital numbers and group orders of Identity, C ₙ, and D ₙ under the same n and k. The reference page has built-in formula shortcuts for Burnside, P ó lya, and common groups.