A000723 -id:A000723 - cp's OEIS frontend

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.

2, 6, 1680, 108972864000, 137047310902965380295426048000000, 5507245320567889066989296412116383715402149139520190633628554443368693760000000000000

Offset: 1

Frank M Jackson, Aug 13 2018

nonn

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.

Frank M Jackson, Table of n, a(n) for n = 1..8
Wikipedia, Cross-polytope

Cf. A000723, A002866, A097801, A198861, A317978.

Mathematica
```
a[n_]:=(2^n)!/(2^(n-1)*n!); Array[a,10]
```

a(n) = (2^n)!/(2^(n-1)*n!) = (2^n)!/A002866(n).

a(n) = 2 * A000723(n). - Alois P. Heinz, Aug 15 2018

cp's OEIS Frontend

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.

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