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A317251
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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.
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1
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OFFSET
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1,1
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COMMENTS
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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|.
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.
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LINKS
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FORMULA
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a(n) = (2^n)!/(2^(n-1)*n!) = (2^n)!/A002866(n).
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MATHEMATICA
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a[n_]:=(2^n)!/(2^(n-1)*n!); Array[a, 10]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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