Row-column collision lemma ($\lambda \ge \mu ⟹\text{collision or }(\lambda =\mu ,u=rct)$)

If the rows of a Young tableau of shape $\lambda$ have no repeated intersection with the columns of one of shape $\mu$, then ${\sum }_{i\le k}{\lambda }_i\le {\sum }_{i\le k}{\mu }_i$ for every $k$. If also $\lambda \ge \mu$ in dictionary order on integer partitions, the shapes are equal. Saturation of the prefix bounds places one entry of each eligible row in each column, giving $u=rct$ with $r\in R(t)$ and $c\in C(t)$. A collision instead gives a transposition that makes row symmetrization followed by the relevant column antisymmetrization vanish.