A combinatorial proof of a partition perimeter inequality

TL;DR

This paper provides a combinatorial proof of a partition perimeter inequality, linking partition statistics to compositions, and extends related perimeter theorems through new combinatorial insights.

Contribution

It offers a novel combinatorial proof of an existing partition perimeter inequality and connects it to composition theory, extending related perimeter theorems.

Findings

Established a combinatorial proof relating partition perimeter to compositions.

Extended perimeter inequalities to new classes of partitions and compositions.

Linked existing theorems to new combinatorial interpretations.

Abstract

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 $m \geq 2$ and $n \geq 1$, there are at least as many partitions with perimeter $n$ and parts $\not \equiv 0 \pmod{m}$ 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.