Question 10
Fix a natural number $n$. A function $f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\}$ is called *regular* if $f(0) = 0$ and $f(i) \in \{i-1, f(i-1), f(f(i-1)), \dots\}$ for every $i = 1, \dots, n$. If, for instance, $n = 3$, then the function $f(0) = f(1) = 0, f(2) = f(3) = 1$ is regular, but the function $f(0) = f(1) = f(2) = 0, f(3) = 1$ is not (in the latter case, $f(3)$ violates the regularity condition). Juku chooses a regular function $f$ and tells Miku for each number $k = 0, 1, \dots, n$, how many different arguments $i$ are there such that $f(i) = k$. Can Miku always determine based on this information which regular function Juku had chosen?