= Solution
A <single-box up-move> from row $j$ to row $i<j$ changes the partial sum through row $h$ by
$$
\sum_{r=1}^h(\mu_r-\lambda_r)=\begin{cases}1&i\leq h<j,\\0&h<i\text{ or }h\geq j.\end{cases}
$$
Every change is nonnegative, so $\lambda\unlhd\mu$ in the <dominance order on partitions>. Transitivity gives the same conclusion along any chain of up-moves. Thus \b[the existence of the indicated chain implies dominance]. A chain of length zero handles equality; each positive-length move must still leave a <partition of an integer>.
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