Abstract
The cycle index polynomial is described for the action of the full isometry group of the n-dimensional hypercube on its q-dimensional cells. This group action is interpreted as \(C_2^n \cdot S_n \) acting on the set of unit basis vectors in Rn and their opposites. A kind of generating function that yields all these polynomials at once is obtained by Mobius inversion. The same technique is applied to the simpler case of the n-dimensional simplex.
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