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%I A317251 #30 May 23 2022 03:51:48
%S A317251 2,6,1680,108972864000,137047310902965380295426048000000,
%T A317251 5507245320567889066989296412116383715402149139520190633628554443368693760000000000000
%N A317251 a(n) is the number of ways to paint the 2^n cells of dimension n-1 that bound a regular convex n-orthoplex polytope using exactly 2^n colors where n is the dimension of Euclidean space.
%C A317251 Let G, the group of rotations in n-dimensional Euclidean space, act on the set of (2^n)! paintings of an n-orthoplex bound by 2^n cells of dimension n-1. There are (2^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 2^(n-1)*n! = A002866(n). So by Burnside's Lemma a(n) = (2^n)!/|G|.
%C A317251 See A198861(3) for the number of ways to paint the octahedron a(3) in the Platonic solids and A317978(3) for the 4-orthoplex a(4) in the regular convex 4-polytopes.
%H A317251 Frank M Jackson, <a href="/A317251/b317251.txt">Table of n, a(n) for n = 1..8</a>
%H A317251 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%F A317251 a(n) = (2^n)!/(2^(n-1)*n!) = (2^n)!/A002866(n).
%F A317251 a(n) = 2 * A000723(n). - _Alois P. Heinz_, Aug 15 2018
%t A317251 a[n_]:=(2^n)!/(2^(n-1)*n!); Array[a,10]
%Y A317251 Cf. A000723, A002866, A097801, A198861, A317978.
%K A317251 nonn
%O A317251 1,1
%A A317251 _Frank M Jackson_, Aug 13 2018

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