Fixed Perimeter Analogues of Several Partition Results Related to Parity

Abstract

In 2016, Straub proved that Euler's classic partition identity holds true for partitions with largest hook (perimeter) $n$. This inspired further study of the relationship between classical partitions and fixed perimeter partitions. We extend the study of parity bias inequalities, first introduced by Kim, Kim, and Lovejoy in 2020, to the fixed perimeter setting and show using combinatorial methods that fixed perimeter analogues of many classical parity bias results can be proven and generalized. We also extend these methods to prove similar inequalities for the fixed perimeter analogues of PED and POD partitions. We additionally develop recursive formulas for the number of perimeter $n$ partitions with odd parts distinct and even parts unrestricted and with even parts distinct and odd parts unrestricted.