关于固定分拆周长的一个猜想的证明
A Proof of a Conjecture on Fixed Perimeter Partitions
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中文总结 AI 辅助
本文证实了Gray等人提出的对所有n≥9成立的奇偶偏差不等式的固定周长类似猜想，该猜想源于分拆理论的固定周长相关研究。
中文摘要 AI 辅助
近年来，寻找各类分拆理论恒等式与不等式的固定周长类似问题已成为活跃的研究领域。Gray、Payne、Swisher 和 Watson（《离散数学》，2026）建立了若干受欧拉著名分拆恒等式启发的分拆理论结果的固定周长类似结果。最近，他们在另一篇论文（arXiv:2608.00421，2026）中探讨了与奇偶偏差相关的不等式的固定周长类似结果。Kim、Kim 和 Lovejoy（《欧洲组合学杂志》，2020）引入奇偶偏差概念后，提出猜想：对所有n≥20，pdₒ(n)>pdₑ(n)，其中pdₒ(n)表示n的不同分拆中奇数部分数量多于偶数部分数量的分拆数，pdₑ(n)则表示偶数部分数量多于奇数部分数量的分拆数。作者与Banerjee、Bhattacharjee、Dastidar和Saikia（《欧洲组合学杂志》，2022）证明了该猜想。Gray等人猜想该不等式的固定周长类似结果对所有n≥9成立，本文证实了他们的这一猜想。
英文摘要
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.
