Given a permutation's lexicographic number, is it possible to get any item in it in O(1)

Question

I want to know whether the task explained below is even theoretically possible, and if so how I could do it.

You are given a space of N elements (i.e. all

Answer 1

Your question is a bit moot, because your input size for an arbitrary permutation index has size log(N!) (assuming you want to represent all possible permutations) which is Theta(N log N), so if N is really large then just reading the input of the permutation index would take too long, certainly much longer than O(1). It may be possible to store the permutation index in such a way that if you already had it stored, then you could access elements in O(1) time. But probably any such method would be equivalent to just storing the permutation in contiguous memory (which also has Theta(N log N) size), and if you store the permutation directly in memory then the question becomes trivial assuming you can do O(1) memory access. (However you still need to account for the size of the bit encoding of the element, which is O(log N)).

In the spirit of your encryption analogy, perhaps you should specify a small SUBSET of permutations according to some property, and ask if O(1) or O(log N) element access is possible for that small subset.