Problem #WSP-000415

Problem

Take a (finite) set $S$, say $[n]$ and a random function $f:S\to S$. What’s the distribution of the limiting size of the image of the iterates of $f$?

That is, $\underset{N\to \mathrm{\infty }}{lim}|f^N([n])|$

By random, let $i\in [n]$. Each $f(i)$ is independently and identically distributed as uniform random variables on $[n]$. One can also think of it as $f$ is taken uniformly from the $n^n$ possible functions $[n]\to [n]$.