Fix two numbers $n,k\in \mathbb{N}$ and abbreviate $[n]:= \{0,1,\dots,n\}$ (as linearly ordered set). Consider diagrams $(I,\varphi)$ of the form $$[n]\supset I\xrightarrow{\varphi}[k]$$ where $\varphi$ is surjective and weakly monotone. Denote by $A(n,k)$ the set of such diagrams. An easy inspection shows that $$|A(n,k)|= \sum_{i=0}^n {n+1\choose i+1}{i\choose k}$$ where the $i$ corresponds to the dimension of $I$ (i.e. $I\cong [i]$).

Now introduce the following relation such diagrams: $(I,\varphi)\leq(I',\varphi')$ if $I\supset I'$ and $\varphi$ is the unique surjective weakly monotone map $I\to[k]$ that extends $\varphi'$. It is clear that $\leq$ is a partial order on $A(n,k)$.

Now my question: what is known about the structure of this partially ordered set and has it appeared in the literature before?

Some specific questions: how many connected components does it have? What are its maximal elements?

Tableau complexes, arXiv:math/0510487. (Imagine your diagram $\left[n\right] \supseteq I \to \left[k\right]$ as a set-valued tableau with a single row of length $k$, where the $i$-th box contains the elements of $I$ that map to the element $i$ of $\left[k\right]$.) $\endgroup$ – darij grinberg Dec 17 '17 at 23:29