AI 中文总结
介绍将讲堂长度引入不同部分分拆，通过设计从奇数部分分拆到不同部分分拆的双射，统一诸多结果，引出相关定理并提出猜想，探讨欧拉分拆定理与讲堂分拆定理的关系。
AI 中文摘要
欧拉分拆定理是整数分拆领域发展的重要一步，吸引了包括西尔维斯特等众多数学家关注，还预示了许多后续结果。分拆理论中的另一个深刻成果是优美的讲堂分拆定理，它是欧拉恒等式的有限版本。本文将讲堂长度引入不同部分的分拆，设计了从奇数部分分拆到不同部分分拆的双射，统一了许多结果，近期关于最小排除数和块指标的各种定理都是此双射的直接推论，还提出了一个有趣猜想。
英文摘要
Euler's partition theorem was an important step in the establishment of Integer Partitions as a field in its own right. It has attracted many great mathematicians' attention, including Sylvester, and has foreshadowed a number of further results. Another deep result in Partition Theory is the beautiful Lecture Hall Partition Theorem, which is a finite version of Euler's identity. In this work, we introduce the lecture hall length to partitions into distinct parts, and devise a bijection from partitions into odd parts to partitions into distinct parts, which unifies many results together. Various recent theorems regarding the minima excludant and block index become immediate consequences of this bijection. We also pose an interesting conjecture which may uncover a deep relation between Euler's partition theorem and lecture hall partition theorem.
