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%I #11 Aug 12 2018 12:44:20
%S 2,210,108972864000,1077167364120207360000
%N The number of ways to paint the cells of the six convex regular 4-polytopes using exactly n colors where n is the number of cells of each 4-polytope.
%C Let G, the group of rotations in 4 dimensional space, act on the set of n! paintings of each convex regular 4-polytopes having n cells. 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 A273509/2. So by Burnside's Lemma a(n)=n!/|G|. a(5) = 120!/7200 and a(6) = 600!/72000 and they are too large to display.
%C See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_4-polytope">Regular 4-polytope</a>
%F a(n) = 2*n!/A273509(n).
%e The second of these six 4-polytopes (in sequence of cell count) is the 4-cube (with 8 cells). It has |G| = 192 rotations with n = 8. Hence a(2) = 8!/192 = 210.
%t {5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}
%Y Cf. A098427, A198861, A273509.
%K nonn,fini
%O 1,1
%A _Frank M Jackson_, Aug 12 2018
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