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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 (list; graph; refs; listen; history; text; internal format)
 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. LINKS Wikipedia, Regular 4-polytope 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 Cf. A098427, A198861, A273509. Sequence in context: A092700 A178388 A056065 * A050445 A167833 A167838 Adjacent sequences:  A317975 A317976 A317977 * A317979 A317980 A317981 KEYWORD nonn,fini AUTHOR Frank M Jackson, Aug 12 2018 STATUS approved

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Last modified May 19 15:03 EDT 2022. Contains 353834 sequences. (Running on oeis4.)