Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs
Matthias Beck, Tristram Bogart, and Tu Pham

TL;DR
This paper systematically enumerates Golomb rulers, proves their count is a quasipolynomial with a reciprocity theorem, and develops an analogue of Stanley's theorem linking chromatic polynomials of mixed graphs to acyclic orientations.
Contribution
It introduces a quasipolynomial enumeration of Golomb rulers and establishes a reciprocity theorem, extending Stanley's theorem to mixed graphs.
Findings
g_m(t) is a quasipolynomial in t
Reciprocity relates counts of rulers with multiplicities
Develops an analogue of Stanley's theorem for mixed graphs
Abstract
A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for \emph{optimal} Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study [g_m(t) := # {\x \in \Z^{m+1} : \, 0 = x_0 < x_1 < ... < x_{m-1} < x_m = t, \text{all} x_j - x_k \text{distinct}},] the number of Golomb rulers with markings and length . Our main result is that is a quasipolynomial in which satisfies a combinatorial reciprocity theorem: equals the number of rulers of length with markings, each counted with its \emph{Golomb multiplicity}, which…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Graph Labeling and Dimension Problems
