Combinatorial Results on Some Nilpotent Subsemigroups of a Semigroup of Order-Decreasing Full Transformations

Let $\mathcal{D}_{n}$ be the semigroup of all order-decreasing full transformations on $X_{n}=\{1,2,\ldots ,n\}$ under its natural order, and let $N(\mathcal{D}_{n})$ be the subsemigroup of all nilpotent elements of $\mathcal{D}_{n}$, where $n\in \mathbb{Z}^+$, the set of all positive integers. In this paper, for $1\leq r\leq n-1$, we determine the cardinality and rank of nilpotent subsemigroup $N(\mathcal{D}_{n,r}) =\{ \alpha\in N(\mathcal{D}_{n}) : \lvert\text{im}(\alpha)\rvert \leq r\}$ of $N(\mathcal{D}_{n})$. We then find the cardinalities of $\mathcal{D}_{n}^{2,2}$ and $N(\mathcal{D}_{n})^{p,p}$. Furthermore, we presents an alternative combinatorial approach to determine the cardinality and rank of ${\mathcal{D}}_{n}(\xi)=\{\alpha \in {\mathcal{D}}_{n}:\alpha ^k=\xi, \text{ for some } k\in {\mathbb {Z}}^{+}\}$, for all idempotent $\xi\in \mathcal{D}_{n}$ within the scope of this study. Here, for all $\alpha\in\mathcal{D}_{n}$, $\text{im}^{c}(\alpha)=\{t\in \text{im}(\alpha): \lvert t\alpha^{-1}\rvert\geq 2\}$. Besides, for all $2\leq p\leq r\leq n$ and $\mathcal{C}\in \{N(\mathcal{D}_n),\mathcal{D}_n\}$, $\mathcal{C}^{p}=\left\{\alpha\in \mathcal{C}: t\in \text{im}^{c}(\alpha) \text{ and } \lvert t\alpha^{-1}\rvert=p\right\}$ and $\mathcal{C}^{p,r}=\left\{\alpha\in \mathcal{C}^p:\bigg\lvert \bigcup\limits_{t\in \text{im}^{c}(\alpha)} t\alpha^{-1}\bigg\rvert=r\right\}$.