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The number of ways to paint the faces of the five Platonic solids using exactly n colors where n is the number of faces of each solid.
3

%I #18 Aug 25 2014 04:43:34

%S 2,30,1680,7983360,40548366802944000

%N The number of ways to paint the faces of the five Platonic solids using exactly n colors where n is the number of faces of each solid.

%C Let G, the group of rotations in 3 dimensional space act on the set of n! paintings of each Platonic solid having n faces. There are n! fixed points in the action table since the only element in G that leaves a painting fixed is the identity element. The order of G is A098427/2. So by Burnside's Lemma a(n)=n!/|G|.

%H David Broughton's Puzzles & Programs, <a href="http://www.iwpcug.org/davidbro/puz0807.htm">Colouring The Platonic Solids</a>

%F a(n) = A053016(n)!/(2*A063722(n)) (see link). - _Michel Marcus_, Aug 24 2014

%o (PARI) lista() = {ve = [6, 12, 12, 30, 30 ]; vf = [4, 6, 8, 12, 20 ]; for (i=1, 5, nb = vf[i]!/(2*ve[i]); print1(nb, ", "););} \\ _Michel Marcus_, Aug 25 2014

%Y Cf. A053016 (number of faces), A063722 (number of edges).

%K nonn,fini,full

%O 1,1

%A _Geoffrey Critzer_, Oct 30 2011