A follow-up to MSE4856267, some context is here.
Consider binary (0/1) sequences of length $n$. Permutations (symmetric group $S_n$ ) act on any sequences - just permuting the symbols. Moreover there are obvious orbits - sequences containing precisely $k$ symbols $1$ - are orbits for any $k$.
That action satisfies the following obvious properties: it is bijective and the sequence with $k$ symbols $1$ are mapped again to the sequence with such property.
However not any map satisfying the properties above is induced by permutations - for example $NOT$ is not induced - see MSE4856267
Question: is there any criteria which distinguishes permutation induced maps from the other ? What can be the algorithm to find such permutation if we have is description as MAP (i.e. pairs of input/output) on the sequences of length $n$ containing precisely $k$ symbols $1$ ?
