Probabilistic results for monoids of order-preserving transformations

Let $\mathcal{PO}_n$ 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 $\mathcal{POI}_n$ denote its submonoids of order-preserving full and injective partial transformations, respectively. For each transformation $\alpha\in\mathcal{PO}_n$, write the random variables $Y(\alpha)=|{\im}\alpha|$ and $Y_r(\alpha)=|{\im}\alpha|$ given that $|{\dom}\alpha|=r$ for $0 \leqslant r \leqslant n$. We determine the probability distribution, expectation and variance of $Y_r$ and $Y$ for $\mathcal{PO}_n$ and $\mathcal{POI}_n$. In particular, $Y_r(\alpha)$ follows a hypergeometric distribution $H(n+r-1,n,r)$ for $\alpha \in \mathcal{PO}_n$, while $Y_r(\alpha)$ is degenerate and $Y(\alpha)$ follows a hypergeometric distribution $H(2n,n,n)$ for $\alpha \in \mathcal{POI}_n$.