Fixed points of orientation-preserving full transformation

Let ${\mathcal{O}\mathcal{P}}_n$ be the monoid of all orientation-preserving full transformations on $X_n=\{1,\dots ,n\}$ with the natural order. For $\alpha \in {\mathcal{O}\mathcal{P}}_n$, let $F(\alpha )=\{y\in X_n:y\alpha =y\}$ and $F(n,m)=\mathrm{\mid }\{\alpha :\mathrm{\mid }F(\alpha )\mathrm{\mid }=m\}\mathrm{\mid }$. Umar posed the question about the number $F(n,m)$ of elements of ${\mathcal{O}\mathcal{P}}_n$ with $m$ fixed points. In this paper, we show that the number $F(n,m)$ of ${\mathcal{O}\mathcal{P}}_n$ is $\left(\genfrac{}{}{0px}{}{2n}{n-m}\right)$ for $2⩽m⩽n$ and get the expectation and probability distribution of the cardinality of fixed-point set $F(\alpha )$ for $\alpha \in {\mathcal{O}\mathcal{P}}_n$.