Fixed points of orientation-preserving full transformation

Abstract

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 $\hat{I}\pm \in {\mathcal{O}\mathcal{P}}_n$, let $F(\hat{I}\pm )=\{y\in X_n:y\hat{I}\pm =y\}$ and $F(n,m)=|\{\hat{I}\pm :|F(\hat{I}\pm )|=m\}|$. 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 $\left(\genfrac{}{}{0ex}{}{2n}{n-m}\right)$ for $2⩽m⩽n$ and get the expectation and probability distribution of the cardinality of fixed-point set $F(\hat{I}\pm )$ for $\hat{I}\pm \in {\mathcal{O}\mathcal{P}}_n$.