Let's say we have a list of colored beads, e.g. [blue, red, red, yellow, blue, red, yellow], and a group G that acts on the list by permuting its elements. Is there an efficient way to calculate the number of distinct permutations up to coloring under this group G?
For example, if G was the symmetric group S7, then the number of permutations is 7!/(2!3!2!) = 210. If G was the alternating group A7, then since we have at least two beads of the same color, we also get 210 (just place the blue and red beads where you want them, which forces the placement of the yellow beads but since they are both yellow we don't lose any distinguishable permutations).
By efficient, I mean a method that doesn't need to iterate over all elements of G. We can assume G is finite but may be very large. The application I have in mind is for permutation puzzles, where if we assume each piece is distinguishable, then there are efficient methods (like Schreier-Sims) to compute the number of different scrambles given the generators for the group. But in many puzzles, pieces are grouped into like colors, so the effective number of scrambles is actually less than this, e.g. the Hungarian rings puzzle.
