A combinatorial proof of a partition perimeter inequality

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher’s theorem: for all $m\ge 2$ and $n\ge 1$, there are at least as many partitions with perimeter $n$ and parts $\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}m)$ as partitions with perimeter $n$ and parts repeating fewer than $m$ times. In this work, we provide a combinatorial proof of their theorem by relating the combinatorics of the partition perimeter to that of compositions. Using this technique, we also show that a composition theorem of Huang implies a refinement of another perimeter theorem of Fu and Tang.