On the ranks of certain subsemigroups of finite orientation-preserving and order-decreasing partial transformations

Let \({\mathscr {P}\mathscr {O}\mathscr {P}\mathscr {D}}_{n}\) ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{D}}_n$ be the semigroup consisting of all orientation-preserving and order-decreasing partial transformations on the finite chain \(X_{n}=\{ 1< \cdots < n \}\) $X_n=\{1<\cdots <n\}$ , and let \({\mathscr {P}\mathscr {O}\mathscr {P}\mathscr {D}}(n,r)=\{ \alpha \in {\mathscr {P}\mathscr {O}\mathscr {P}\mathscr {D}}_{n}: |\textrm{im}\, (\alpha )|\le r\}\) $\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{D}(n,r)=\{\alpha \in {\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{D}}_n:|\text{im}(\alpha )|\le r\}$ for \(1\le r\le n-1\) $1\le r\le n-1$ . We prove that the rank and the idempotent rank of \({\mathscr {P}\mathscr {O}\mathscr {P}\mathscr {D}}(n,r)\) $\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{D}(n,r)$ are both equal to \(\sum \limits _{s=r}^{n} \left( {\begin{array}{c}n\\ s\end{array}}\right) \left( {\begin{array}{c}s\\ r\end{array}}\right) +\frac{(2n-r-1)(r-2)}{2} \) $\sum _{s=r}^n\left(\begin{array}{c}n\\ s\end{array}\right)\left(\begin{array}{c}s\\ r\end{array}\right)+\frac{(2n-r-1)(r-2)}{2}$ for \(1\le r\le n-1\) $1\le r\le n-1$ . Then we conclude that the rank and the idempotent rank of \({\mathscr {P}\mathscr {O}\mathscr {P}\mathscr {D}}_{n}\) ${\mathcal{P}\mathcal{O}\mathcal{P}\mathcal{D}}_n$ are both equal to \(\frac{n^{2}+n+2}{2}\) $\frac{n^2+n+2}{2}$ .