Consider a rectangular $(m \times n)$ matrix $\underline E\_1$ with $m < n$ that has only $0$ or $1$ entries. It has exactly one $1$ entry in each row and not more than one $1$ entry in each column. Consider it being a selection of $m$ rows out of a $(n \times n)$ permutation matrix $\underline P$.


Given $\underline E\_1$ I'm looking for an elegant way to describe the set $\mathcal{P}$ of $((n-m) \times n)$ matrices so that any $\underline E\_2 \in \mathcal{P}$ combined with $\underline E\_1$, like so


$\underline E = \begin{pmatrix} \underline E\_1 \\\ \underline E\_2 \end{pmatrix}$


forms a valid $(n \times n)$ permutation matrix $\underline E$, i.e. something like $\mathcal{P} = \operatorname{percomp}\_n( \underline E\_1 )$ (there are $(n-m)!$ elements in $\mathcal{P}$)


Is there anything like this used in mathematical parlance already?



Complement to part of a permutation matrix
You get all possible $E\_2$ by starting with an $(n-m) \times (n-m)$ permutation matrix, and expand it to being $(n-m) \times n$ by inserting columns of zeros under each of the ones of $E\_1$. That is, there's an easy bijection between your set and the order $(n-m)$ permutation matrices. I think you knew this already. 


I guess what I'm driving at is that even if there is some canonical name for this object in some field of mathematics, the set that you want to name is simple enough that you can describe it in one sentence. If you need to give this set a name, just call it $percomp\_n(E\_1)$, like you wanted to do. People do this all the time. Indeed, most of us would be even *more* confused if you dig up an obscure, but technically correct, name. 


