PolyaCounter

Polya Counter is a professional tool application for learning combinatorial mathematics and counting theory, focusing on Burnside lemma and P ó lya enumeration theorem to help users understand "how many fundamentally different coloring schemes there are under the action of symmetry groups". The application supports setting the number of vertices n, color number k, and group action types (ordinary group {e}, cyclic group C ₙ, dihedral group D ₙ) for regular n-sided shapes. In the Burnside Lemma module, you can gradually view the cyclic structure and number of fixed points of each group element, and calculate the orbital number | X/G |; The Cycle Index module constructs a cycle index polynomial Z_G and supports isochromatic and weighted Polya substitution. Group Comparison can horizontally compare the number of tracks under different groups. At the exploratory level, Polya Enumeration will fully enumerate equivalence classes and support filtering and sorting; Orbit Visualizer displays each track representing shading in a polygonal diagram; Coloring Lab supports interactive coloring of vertices, manual input of schemes, rotation/reflection transformations, and canonical form calculations; Permutation Inspector provides permutation of one line markers, loop markers, and circle length decomposition.