Why does the Gale-Ryser Theorem work? "If a and b are partitions of a positive integer, then there
exists a binary matrix with row sums a and column sums b if and only if a is dominated by the conjugate of b."
This doesn't really seem very intuitive to me, and the proof I've found seems to be a bit too complicated for me to understand particularly well.
Does anyone have some intuition as to why this is the case?
Thank you

It's easy to explain why the condition is necessary. The sum of the first $k$ rows is $a_1 + a_2 + \dots + a_k$. However, the $i^{\text{th}}$ column can only contain at most $\min\{b_i, k\}$ $1$'s in its first $k$ entries. So the sum of the first $k$ rows can be at most $\min\{b_1,k\} + \min\{b_2, k\} + \dots + \min\{b_n,k\}$. This gives us the condition $$\sum_{j=1}^k a_j \le \sum_{i=1}^n \min\{b_i, k\}$$ which is exactly the Gale-Ryser condition.
We can check this condition for any subset of the rows, but it's most constraining when we pick the $k$ largest rows for some $k$, which is why we check this condition with $\mathbf a$ sorted in decreasing order.
So the Gale-Ryser theorem is one of those "the obvious necessary condition is also sufficient" results, where by "obvious" we really only mean that proving it's necessary is much easier than proving the whole theorem.
