On images and pre-images in a graph of the composition of independent uniform random mappings

We study the probability characteristics of the random mapping graph $ f_{\left[k\right]} $ — the composition of $k$ independent equiprobable random mappings $ f_1, \ldots, f_k $, where $f_i\colon \left\{1,\ldots,n\right\}\to \left\{1,\ldots,n\right\}$, $n,k\in\mathbb{N}$, $i=1,\ldots,n$. The following results are obtained. For any fixed $x,y\in S=\{1,\ldots,n\}$, $x\ne y$,
\begin{equation*} \mathbf{P}\{f_{\left[k\right]}(x)=f_{\left[k\right]}(y)\}=\textstyle\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon\\2\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(2,s_{1})}{n^{s_{k-1}-1}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1}), \end{equation*}
where $q(a,b)=\text{C}_{n}^{n-b} \left(\dfrac{b}{n}\right)^a \sum\limits_{l=0}^{b}\text{C}_{b}^l(-1)^l\left(1-\dfrac{l}{b}\right)^a$. For any fixed $x\in S$,
\begin{gather*} \mathbf{P}\{ x\in f_{\left[k\right]}(S)\}=\frac1{n}\textstyle\sum\limits_{l=1}^{n}{\left(\dfrac{(n)_l}{n^l} \right)^k}+\\ +\textstyle\sum\limits_{l=1}^{n-2}\sum\limits_{t=1}^{n-l-1}\sum\limits_{m=1}^{n-t-l}(-1)^{m-1}\text{C}_{n-1}^m\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon\\m\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(m,s_{1})}{n^{s_{k-1}}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1})V^{\left\{k,m\right\}}_{s_1,\ldots,s_{k-1}}, \end{gather*}
where
\begin{gather*} V^{\left\{k,m\right\}}_{s_1,\ldots,s_{k-1}}=\mathbf{P}\{x\in H_{f_{\left[k\right]}}^{\left(t,l\right)}\bigm| D^{\left\{k\right\}}_{s_1,\ldots,s_{k-1},1}\left(y_1,\ldots,y_m\right),f_{\left[k\right]}\left(y_1\right)=x \}=\\ =\frac{1}{n}\textstyle\prod\limits_{i=m+1}^{t+l+m-1}{\left( 1-\dfrac{i}{n} \right)}\prod\limits_{i=1}^{k-1}\prod\limits_{j=s_i+1}^{t+l+s_i-2}{\left( 1-\dfrac{j}{n} \right)}\sum\limits_{v=0}^{k-1}\prod\limits_{u=1}^{v}{\left( 1-\dfrac{t+l+s_u-1}{n} \right)}, \end{gather*}
$H_f^{\left(t,l\right)}$ is $t$-th layer of cycles of length $l$ in graph $G_f$, $D^{\left\{k\right\}}_{s_1,\ldots,s_{k}}(y_1,\ldots,y_m)=\textstyle\bigcap\limits_{i=1}^{k} \{|\{f_{\left[i\right]}(y_1),\ldots,f_{\left[i\right]}(y_m)\}|=s_i\}$, and $(n)_z=n(n-1)\dots(n-z+1)$. For any fixed $x\in S\setminus S'$ and for any $r\in \{1,\ldots,n-1\}$, $S'\subseteq S$, $|S'|=r$, $z\in \{1,\ldots,n\}$,
\begin{gather*} \mathbf{P}\{\tau_{f_{\left[k\right]}}(x)=z,\mathcal{R}_{f_{\left[k\right]}}(x)\cap S'=\varnothing \}=\\ =\left(1-\left(1-\frac{z}{n}\right)\left( 1-\frac{z-1}{n} \right)^{k-1}\right)\left(\frac{\left(n\right)_{z-1}}{n^{z-1}} \right)^{k-1}\frac{\left(n\right)_{r+z}}{n^{z-1}\left(n\right)_{r+1}}, \end{gather*}
where $\mathcal{R}_{f_{\left[k\right]}}(x)$ is the aperiodicity segment of vertex $x$ in the graph of mapping $f_{\left[k\right]}$, $\tau_{f_{\left[k\right]}}(x)=\min\{ t\in \mathbb{N}\colon {f_{\left[k\right]}}^t(x)\in \{ x,{f_{\left[k\right]}}(x),\dots,{f_{\left[k\right]}}^{t-1}(x) \}\}$. For any fixed $x,y\in S$, $x\ne y$, and for any $r\in\{1,\ldots,n\}$,
\begin{equation*} \mathbf{P}\{y \in (f_{\left[k\right]})^{-r}(x)\}=\frac1n\left(1-\frac1{n-1}\textstyle\sum\limits_{z\in Q_r\setminus\{1\}}\left(\dfrac{(n)_z}{n^z}\right)^k\right), \end{equation*}
where $Q_r=\{m\in \mathbb{N}\colon m|r\}$.