Dominance order on partitions

For partitions of the same integer, $\lambda$ dominates $\mu$ whenfor every $r$.

A single-box up-move changes a partition of an integer by adding one to row $i$ and subtracting one from row $j>i$, provided the result remains a partition. It increases exactly the partial sums ending between rows $i$ and $j-1$. A partition $\mu$ dominates $\lambda$ precisely when it is obtainable from $\lambda$ by a sequence of these moves: fill the first deficient row from the row at the first return of the cumulative deficit to zero. This preserves the partition inequalities and decreases the distance to $\mu$.