Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix?
Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we determine if there's a $5 \times 5$ matrix whose rows and column sums both give rise to that partition?
In the example, it's realized by this matrix: $$\begin{bmatrix} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ but I cheated: I generated the example partition from the matrix.
Observations:
The obvious necessary condition is that $\max P \leq |P|$.
This is equivalent to asking when is $P$ the out-degree sequence of a directed graph where any vertex may have one loop. (So the Erdős–Gallai theorem or Havel–Hakimi algorithm could be applied in some cases, replacing each undirected edge with a bidirected edge; in fact, they would work in my above example.)
If we just brute force started filling in $1$'s, we can get stuck. Here I proceed row-by-row, placing a $1$ wherever possible: $$ \begin{array}{c|ccccc} & 4 & 3 & 3 & 2 & 2 \\ \hline 4 & 1 & 1 & 1 & 1 & 0 \\ 3 & 1 & 1 & 1 & 0 & 0 \\ 3 & 1 & 1 & 1 & 0 & 0 \\ 2 & 1 & 0 & 0 & 1 & 0 \\ 2 & 0 & 0 & 0 & 0 & ??? \\ \end{array} $$
Motivation:
This is a first step in an attempt at answering my Mathoverflow question: What is the minimum number of filled cells in a partial Latin rectangle with autotopism group $\cong C_2$ and autoparatopism group $\cong S_3$?. An answer to this question might help reduce the search space.