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.

tetrahedron: (x1^4+3x2^2+8x1*x3)/12 cube: (x1^6+6x2^3+3x1^2*x2^2+8x3^2+6x1^2*x4)/24 octahedron: (x1^8+9x2^4+8x1^2*x3^2+6x4^2)/24 dodecahedron: (x1^12+15x2^6+20x3^4+24x1^2*x5^2)/60 icosahedron: (x1^20+15x2^10+20x1^2*x3^6+24x5^4)/60 tesseract: (32x6^4+x2^12+48x8^3+x1^24+24x1^2*x2^11+12x2^2*x4^5+32x3^8+12x4^6

  • 18x1^4*x2^10+12x1^4*x4^5)/192

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