  Starting from billiard partitions which arose recently in the description of
periodic trajectories of ellipsoidal billiards in $d$-dimensional Euclidean
space, we introduce a new type of separable integer partition classes, called
type B. We study the numbers of basis partitions with $d$ parts and relate them
to the Fibonacci sequence and its natural generalizations. Remarkably, the
generating series of basis partitions can be related to the quiver generating
series of symmetric quivers corresponding to the framed unknot via
knots-quivers correspondence, and to the count of Schr\"oder paths.
