  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.
