cp's OEIS Frontend

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.

%I A317978 #19 Jun 17 2025 00:40:34
%S A317978 2,210,108972864000,1077167364120207360000
%N 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.
%C A317978 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 A317978 See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.
%C A317978 a(6) = 17577...66368*10^146 has 1405 digits. - _Georg Fischer_, Jun 16 2025
%H A317978 Georg Fischer, <a href="/A317978/b317978.txt">Table of n, a(n) for n = 1..5</a>
%H A317978 Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_4-polytope">Regular 4-polytope</a>
%F A317978 a(n) = 2*A063924(n)! / A273509(n). [Corrected by _Georg Fischer_, Jun 16 2025]
%e A317978 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.
%p A317978 {5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200};
%t A317978 {5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}
%Y A317978 Cf. A063924, A098427, A198861, A273509.
%K A317978 nonn,fini
%O A317978 1,1
%A A317978 _Frank M Jackson_, Aug 12 2018