Open Access

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

Research in Number Theory20162:2

DOI: 10.1007/s40993-015-0033-3

Received: 29 July 2015

Accepted: 14 December 2015

Published: 1 March 2016

Abstract

The 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.