A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

TL;DR

This paper provides a geometric, bijective proof that the generating function for partitions with bounded differences is rational, using polyhedral cone models to establish a combinatorial tiling and bijection.

Contribution

It introduces a novel geometric approach modeling partitions as lattice points in polyhedral cones, proving rationality through a bijective, combinatorial tiling method.

Findings

Generating function for bounded difference partitions is rational.

Polyhedral cone model tiles a single simplicial cone.

Bijection offers a combinatorial interpretation of the generating function.

Abstract

The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.