Fixed points of orientation-preserving full transformation

TL;DR

This paper determines the exact count of orientation-preserving full transformations with a given number of fixed points and analyzes their fixed-point set distribution.

Contribution

It provides a closed-form formula for the number of such transformations with specified fixed points and explores their fixed-point set distribution.

Findings

Number of transformations with m fixed points is inom{2n}{n-m} for 2 m m  n.

Derived the expectation of the fixed-point set size.

Established the probability distribution of fixed-point set cardinality.

Abstract

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