Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($\forall x$, $\alpha(x)\le x$).
Let N($\mathcal{C_n})$ be all nilpotent elements of $\mathcal{C}_n$, i.e., those $\alpha\in\mathcal{C}_n$ such that some power of $\alpha$ equals the absorbing element (the constant function $1$.)
We say $a(\in\mathcal{C}_n$) $r$-nilpotent if $r$ is the smallest positive integer satisfying $a^r=0$. Then I guess that there is a well-defined bijection
$a\mathcal{C}_na\to N(\mathcal{C}_r)$.
Although I have tried many times, I could not construct it.