An Ehrhart theoretic approach to generalized Golomb rulers

TL;DR

This paper extends Ehrhart theoretic methods to analyze generalized Golomb rulers, providing a unified geometric framework for counting and structural properties of these combinatorial objects.

Contribution

It generalizes the inside-out polytope approach to various types of generalized Golomb rulers, broadening the scope of geometric combinatorics in this area.

Findings

Unified geometric framework for counting generalized Golomb rulers

Structural results for various generalizations

Extension of Ehrhart theory to new combinatorial objects

Abstract

A Golomb ruler is a sequence of integers whose pairwise differences, or equivalently pairwise sums, are all distinct. This definition has been generalized in various ways to allow for sums of h integers, or to allow up to g repetitions of a given sum or difference. Beck, Bogart, and Pham applied the theory of inside-out polytopes of Beck and Zaslavsky to prove structural results about the counting functions of Golomb rulers. We extend their approach to the various types of generalized Golomb rulers.