# Combinatorics of periodic ellipsoidal billiards

## TL;DR

This paper explores the combinatorial structures of billiard partitions related to periodic trajectories in ellipsoidal billiards across Euclidean and pseudo-Euclidean spaces, introducing weighted partitions and their generating functions.

## Contribution

It introduces weighted partitions to encode caustic types and derives closed-form generating functions for these partitions.

## Key findings

- Characterization of billiard partitions for periodic trajectories
- Introduction of weighted partitions for caustic types
- Closed-form generating functions for these partitions

## Abstract

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. We provide closed forms for the generating functions of these partitions.