A Proof of a Conjecture on Fixed Perimeter Partitions

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 $pd_o(n)>pd_e(n)$ for all $n\ge 20$, where $pd_o(n)$ (respectively, $pd_e(n)$) 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.