Combinatorics of periodic ellipsoidal billiards

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