We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.

中文翻译：

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周期性椭球台球的组合

我们研究在d维欧几里德空间和拟欧几里德空间中椭圆形台球的周期性轨迹描述中最近出现的台球隔板的组合。这样的分区唯一地将苛性碱集合编码，直至其类型，从而产生周期性轨迹。周期轨迹的周期是最大的部分，而绕组数是相应分区的剩余求和。为了也考虑腐蚀性因素的类型，我们引入了加权分区，并为这些分区的生成函数提供了封闭形式。