Probabilistic results for monoids of order-preserving transformations

Abstract

Let ${\mathcal{P}\mathcal{O}}_n$ be the monoid of all order-preserving partial transformations on $X_n=\{1,\dots ,n\}$ with the natural order, and let ${\mathcal{O}}_n$ and ${\mathcal{P}\mathcal{O}\mathcal{I}}_n$ denote its submonoids of order-preserving full and injective partial transformations, respectively. For each transformation $\hat{I}\pm \in {\mathcal{P}\mathcal{O}}_n$, write the random variables $Y(α)=|{\im}α|$ and $Y_r(α)=|{\im}α|$ given that $|{\dom}α|=r$ for $0⩽r⩽n$. We determine the probability distribution, expectation and variance of $Y_r$ and $Y$ for ${\mathcal{P}\mathcal{O}}_n$ and ${\mathcal{P}\mathcal{O}\mathcal{I}}_n$. In particular, $Y_r(\hat{I}\pm )$ follows a hypergeometric distribution $H(n+r-1,n,r)$ for $\hat{I}\pm \in {\mathcal{P}\mathcal{O}}_n$, while $Y_r(\hat{I}\pm )$ is degenerate and $Y(\hat{I}\pm )$ follows a hypergeometric distribution $H(2n,n,n)$ for $\hat{I}\pm \in {\mathcal{P}\mathcal{O}\mathcal{I}}_n$.