Relaxed complete partitions: an error-correcting Bachet's problem

TL;DR

This paper introduces and classifies relaxed complete partitions inspired by an error-correcting extension of Bachet's weights problem, using polyhedral geometry and Brion's formula for enumeration.

Contribution

It provides a new framework for understanding relaxed complete partitions through lattice points in polyhedra and generalizes previous classifications and enumerations.

Findings

Characterization of relaxed complete partitions via lattice points in polyhedra

Enumeration of minimal such partitions using Brion's formula

Extension of prior work on complete partitions

Abstract

Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of R{\o}dseth on enumerating minimal complete partitions.