Linked partition ideals and Euclidean billiard partitions

TL;DR

This paper introduces a new framework for Euclidean billiard partitions, deriving generating function identities that confirm previous results on these special integer partitions related to billiard trajectories.

Contribution

It refines linked partition ideals to establish generating function identities for Euclidean billiard partitions, connecting combinatorics with billiard trajectory studies.

Findings

Derived trivariate generating function identities.

Confirmed properties of Euclidean billiard partitions.

Linked partition ideals framework applied to billiard partitions.

Abstract

Euclidean billiard partitions were recently introduced by Andrews, Dragovic and Radnovic in their study of periodic trajectories of ellipsoidal billiards in the Euclidean space. They are integer partitions into distinct parts such that (E1) adjacent parts are never both odd; (E2) the smallest part is even. By refining the framework of linked partition ideals, we establish a couple of relevant trivariate generating function identities, from which the result of Andrews, Dragovic and Radnovic follows as an immediate consequence.