1. Why is it important that they are integers? (Any reason you cannot operate on sorted $p_i$?) 2. It is important that it is Shannon, not some other entropy, e.g. a function of $\sum p_i^2$?

@Memming An utterance to express that here it is awkward to use information in another, informal meaning.

It seems I will stick to the partial inverse, as it is in use (see the other answer), and self-explanatory. Accept for lining to a review.

@djechlin It's called growth hacking. Sometimes efficient, but once overdone in can backfire. Indeed, RG got a lot of users. But "I have yet to hear any positive success story from my peers". For LinkedIn I know a lot of success stories.

It is used in optics, just with no name (as often it is used only once). My paper bases on jumping back and forth (e.g. unitary matrix -> partial inverse -> diagonalization -> partial inverse of $C$, $D$ and $C^{-1}$). In any case, all other names than partial inverse seem to me a bit contrived (i.e. there is no way I would have guessed what do they mean).

So, as I see, it has many names. Thanks for this review!

Just curious, why inversion not inverse?

I am not sure if I understand your question. You know about Tsallis and Rényi entropies - so what is the exact question "1."? As you know, Rényi axiomatically defines entropy and arrives at a broader class than Shannon entropy. BTW: it is used, see e.g. q-Gaussian.

@TobiasFritz Hi! I would be grateful for sending them to me. I wasn't working on it in the meantime.

The numbers you described are know oeis.org/A000140. But what is the question? (As of now it sound open-ended.)