Combinatorial results for zero-divisors regarding right zero elements of order-preserving transformations

For any positive integer $n$, let $\mathcal{O}_{n}$ be the semigroup of all order-preserving full transformations on $X_{n}=\{1<\cdots <n\}$. For any $1\leq k\leq n$, let $\pi_{k}\in \mathcal{O}_{n}$ be the constant map defined by $x\pi_{k}=k$ for all $x\in X_{n}$. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of $\pi_{k}$: \begin{eqnarray*} \mathsf{L}_{k} &=& \{ \alpha\in \mathcal{O}_{n}:\alpha\beta=\pi_{k} \mbox{ for some }\beta\in \mathcal{O}_{n} \setminus\{\pi_{k}\} \}, \mathsf{R}_{k} &=& \{ \alpha\in \mathcal{O}_{n}:\gamma\alpha=\pi_{k} \mbox{ for some }\ \gamma\in \mathcal{O}_{n}\setminus\{\pi_{k}\} \}, \ \mbox{and} \ \mathsf{Z}_{k}=\mathsf{L}_{k}\cap \mathsf{R}_{k}. \end{eqnarray*} We determine the structures and cardinalities of $\mathsf{L}_{k}$, $\mathsf{R}_{k}$ and $\mathsf{Z}_{k}$ for each $1\leq k\leq n$. Furthermore, we compute the ranks of $\mathsf{R}_{1}$,\, $\mathsf{R}_{n}$,\, $\mathsf{Z}_{1}$,\, $\mathsf{Z}_{n}$ and $\mathsf{L}_{k}$ for each $1\leq k\leq n$, because these are significant subsemigroups of $\mathcal{O}_{n}$.