I am interested in functions $f\colon X\to X$ (where $X$ is some countable set) such that for every $x \in X$ there exists a polynomial $P_x$ such that $\#(f^k)^{-1}(x)=P_x(k)$ for all $k \geq 1$.

Specifically:

- Are there any known structural results about such funcitons?
- Do you know of some (algebraic?) context in which functions like this arise?

Why I am interested in such self-maps: at the end of this paper (https://arxiv.org/abs/1708.04850, section 20) we find some such $f$ (specifically, one for each simply-laced root system).