The number of ways to color the vertices of an $n$-dimensional cube can be obtained from the Redfield-Pólya theorem by obtaining the cycle index of the relevant permutations, of which there are $n!2^n$. A method for doing this using conjugacy classes of the $n!$ permutations of the axes leads to quick results. Both the number of permutations in each conjugacy class and the odd and even cycle indices for the $2^n$ axis reversals for a given conjugacy class can be quickly determined. It would seem to me that this method, simple as it is, must have appeared in the literature, but I have been unable to uncover it. Before spending time writing up a proof of this method, perhaps someone could direct me to the work of a predecessor.



Quick enumeration for the coloring of the vertices of an n-dimensional cube
The group of symmetries of a hypercube is known as the [hyperoctahedral group](https://en.wikipedia.org/wiki/Hyperoctahedral_group). For $n$-dimensional hypercube, it is formed by the wreath product of symmetric groups $S\_2$ and $S\_n$. The classic reference for enumeration under actions of wreath products is [Enumeration under two representations of the wreath product](https://projecteuclid.org/euclid.acta/1485889789) by E. M. Palmer and R. W. Robinson.


