This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
What are the cycle polynomials for the Platonic solids?
For future reference, here are the cycle polynomials for the five platonic
solids (and I threw in the tesseract for good measure). Most combinatoric
coloring problems are simple plug-ins to these polynomials. For details,
see any book on combinatorics that presents Polya counting theory.