AI 中文总结
该研究将奇偶性偏差不等式等分拆相关结果扩展至固定周长场景，推导了两类特殊分拆的递归公式，推广了经典分拆理论的相关结论。
AI 中文摘要
2016年，Straub证明了欧拉经典分拆恒等式对最大钩（即周长）为n的分拆成立，这启发了对经典分拆与固定周长分拆关系的进一步研究。我们将Kim、Kim和Lovejoy于2020年首次提出的奇偶性偏差不等式研究扩展至固定周长场景，通过组合方法证明了许多经典奇偶性偏差结果的固定周长类似物可被证明和推广。我们还将这些方法扩展，以证明PED和POD分拆的固定周长类似物的类似不等式。此外，我们针对两类分拆推导了递归公式：一类是周长为n、奇部互不相同且偶部无限制的分拆；另一类是周长为n、偶部互不相同且奇部无限制的分拆。
英文摘要
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.
Comments17 pages
