This is probably a question that has been asked before and is addressed in a textbook, but I can't find the answer, so I'll ask anyways.
Let $P =\lbrace p_{0}, p_{1}, p_{2}, ... p_{n-1}\rbrace$ be an arbitrary permutation of the set $\lbrace0, 1, 2, ... n-1\rbrace$. So the cardinality of $P$ is $n$ and the count of possible $P$ is $n!$.
Define a "self permuting" function $f: P \mapsto P'$ such that $f(p'_{i})=p_{p_{i}}$. For example, $P= \lbrace3,2,0,1\rbrace, P' = \lbrace1,0,3,2\rbrace$ and $P''= \lbrace 0, 1, 2, 3 \rbrace$. This function probably already has a name somewhere. I just don't know what it is or how to find it. I am trying to determine which $P$ cycles through all $n!$ permutations. There are obvious permutations where it doesn't such, as when a $p_{i}=i$ or a conjugate pair, $p_{i}=p_{j}, \;\; p_{j}= p_{i}$. Outside of brute force, which is computationally impractical for all but the smallest $n$, is there a method to do so?
