Polynomial functional-graph cycle-count conjecture

Let $q$ be a prime power. For $d\in\mathbb{Z}^{\geq 0}$, let $\mathcal{P}(q,d)$ denote the average number of cycles in the functional graphs $\Gamma_f$ of degree-$d$ polynomials $f\in\mathbb{F}_q[x]$, and for $k\in\mathbb{Z}^{>0}$ let $\mathcal{P}(q,d,k)$ denote the average number of $k$-cycles. Let $\mathcal{K}(q)$ be the corresponding main-term function defined in the paper.

Polynomial cycle-count conjecture. For any $k\in\mathbb{Z}^{>0}$ and any $d\in\mathbb{Z}^{\geq 0}$,

$$\mathcal{P}(q,d,k)=\frac{q(q-1)\cdots(q-(k-1))}{kq^k}+O\left(\frac{1}{q}\right),$$

where the implied constant depends only on $d$. In particular, $\mathcal{P}(q,d)=\mathcal{K}(q)+O(1)$.

This is the stronger implication of the paper's heuristic independence assumption for the random variables governing $k$-cycles and polynomial constraints. It predicts the average number of $k$-cycles, uniformly in the stated parameters up to an error of order $1/q$, with dependence of the implied constant only on $d$.