Fixed perimeter analogues of some partition results

TL;DR

This paper investigates fixed perimeter analogues of classical partition identities, extending Euler's identity to partitions with a fixed largest hook (perimeter) and exploring related inequalities.

Contribution

It introduces new fixed perimeter partition results inspired by Euler's identity, expanding the understanding of partition identities in this setting.

Findings

Fixed perimeter analogues of Euler's partition identity are established.

Partition inequalities in the fixed perimeter context are explored.

The work broadens the scope of classical partition results to fixed perimeter scenarios.

Abstract

Euler's partition identity states that the number of partitions of $n$ into odd parts is equal to the number of partitions of $n$ into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook (perimeter) $n$. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.