Billiard Partitions, Fibonacci Sequences, SIP Classes, and Quivers

TL;DR

This paper introduces a new class of integer partitions inspired by billiard trajectories, linking them to Fibonacci sequences, quiver series, and Schröder paths, revealing deep combinatorial and geometric connections.

Contribution

It defines type B billiard partitions, explores their relation to Fibonacci sequences, and connects their generating series to quiver and path counting frameworks.

Findings

Basis partitions with d parts relate to Fibonacci numbers.

Generating series connect to quiver series of symmetric quivers.

Partition counts relate to Schröder path enumeration.

Abstract

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.