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We know that the maximum possible numbers of stereoisomers of a molecule is given by the van't Hoff rule, which says that:

$$Total \space stereoisomers = 2^{n}$$

Where $n$ is the number of stereocenters. However, this is just the theoretical maximum: in fact, if the molecule has a certain number of planes of symmetry, the number of isomers drastically decreases. Is there a formula that accounts for the number of planes of symmetry? The only contribution I could find was here:

How to derive these general formulae for number of stereoisomers of a compound with a possible plane of symmetry?

But this only accounts for one plane of symmetry. What if we have more? Things obviously get more difficult, as not only the number of planes, but also how they divide the molecule will influence how many stereoisomers we'll have. Do you think it would be possible to generalize the formula above anyway?

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