A198861 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.

2, 30, 1680, 7983360, 40548366802944000

Offset: 1

Author

Geoffrey Critzer, Oct 30 2011

Keywords

• nonn
• fini
• full

Comments

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|.

Links

• David Broughton's Puzzles & Programs, Colouring The Platonic Solids

Crossrefs

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

Programs

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

Formula

a(n) = A053016(n)!/(2*A063722(n)) (see link). - Michel Marcus, Aug 24 2014