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A317978
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.
1
2, 210, 108972864000, 1077167364120207360000
OFFSET
1,1
COMMENTS
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.
See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.
FORMULA
a(n) = 2*n!/A273509(n).
EXAMPLE
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.
MATHEMATICA
{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}
CROSSREFS
KEYWORD
nonn,fini
AUTHOR
Frank M Jackson, Aug 12 2018
STATUS
approved