A Proof of a Conjecture on Fixed Perimeter Partitions

Abstract

Finding fixed perimeter analogues of various partition theoretic identities and inequalities has recently emerged as an active area of research. Gray, Payne, Swisher, and Watson [\textit{Discrete Math.}, 2026] established several fixed perimeter analogues of partition theoretic results inspired by Euler's celebrated partition identity. Very recently, in a separate work [\textit{ar{X}iv:2608.00421}, 2026], they explored fixed perimeter analogues of inequalities related to parity biases. Introducing the concept of parity bias, Kim, Kim, and Lovejoy [\textit{Eur. J. Comb.}, 2020] conjectured that $p{d}_o(n)>p{d}_e(n)$ for all $n\ge 20$, where $p{d}_o(n)$ (respectively, $p{d}_e(n)$) denote the number of partitions of $n$ into distinct parts having more odd parts (respectively, even parts) than even parts (respectively, odd parts). The author, together with Banerjee, Bhattacharjee, Dastidar, and Saikia [\textit{Eur. J. Comb.}, 2022], proved this conjecture. Gray, Payne, Swisher, and Watson conjectured that a fixed perimeter analogue of this inequality holds for all $n\ge 9$. In this paper, we confirm their conjecture.