Dictionary order on integer partitions ($\lambda >\mu ⟺{\lambda }_k>{\mu }_k\text{ at the first difference}$)

Pad partitions with trailing zeros and compare their first unequal parts. A partition is larger in dictionary order on integer partitions when that first unequal part is larger. This is a total order, whereas dominance order on partitions compares every prefix sum and is generally partial. The row-column collision lemma links these two orders.