Probabilistic results for monoids of order-preserving transformations

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 $\alpha \in {\mathcal{P}\mathcal{O}}_n$, write the random variables $Y(\alpha )=\mathrm{\mid }\text{im}\alpha \mathrm{\mid }$ and $Y_r(\alpha )=\mathrm{\mid }\text{im}\alpha \mathrm{\mid }$ given that $\mathrm{\mid }\text{dom}\alpha \mathrm{\mid }=r$ 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, $Y_r(\alpha )$ follows a hypergeometric distribution $H(n+r-1,n,r)$ for $\alpha \in {\mathcal{P}\mathcal{O}}_n$, while $Y_r(\alpha )$ is degenerate and $Y(\alpha )$ follows a hypergeometric distribution $H(2n,n,n)$ for $\alpha \in {\mathcal{P}\mathcal{O}\mathcal{I}}_n$.