Franklin恒等式的一个周长类比及与部分奇偶性相关的不等式
A perimeter analogue of Franklin's identity and an inequality related to the parity of parts
- Michigan Technological University（密歇根理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过生成函数与渐近分析，证明了关于分拆周长不等式的两个猜想，并建立了与无峰Motzkin路径的联系。
AI中文摘要：
我们证明了两个关于分拆周长不等式的猜想。Gray、Payne和Watson猜想，如果将分拆的大小替换为其周长，Franklin分拆恒等式最终会成为一个不等式。我们通过相应生成函数的渐近分析证明了这一猜想。Gray、Payne、Swisher和Watson还猜想，对于固定周长的分拆，其奇数部分的数量倾向于多于偶数部分的数量。我们通过推导相应的生成函数证明了这一猜想，并给出了系数的正递推关系。在奇数部分与偶数部分数量相等的情况下，我们提供了与无峰Motzkin路径的联系。
英文摘要：
We prove two conjectures regarding partition perimeter inequalities. It was conjectured by Gray, Payne, and Watson that Franklin's partition identity becomes an eventual inequality if one replaces the size of the partition with its perimeter. We prove this conjecture by asymptotic analysis of the corresponding generating functions. Gray, Payne, Swisher, and Watson also conjectured that there is a bias for partitions with fixed perimeter to have more odd parts than even parts. We prove this conjecture by deriving the corresponding generating functions and give a positive recurrence for the coefficients. In the case that there are an equal number of odd parts and even parts we provide a connection to peakless Motzkin paths.
