Constructing Skolem sequences via generating trees

TL;DR

This paper introduces a generating tree approach to construct all Skolem sequences efficiently by translating the problem into perfect matchings and tracking arc lengths, significantly reducing the search space.

Contribution

The paper presents a novel generating tree method for Skolem sequences, improving enumeration efficiency by translating the problem into perfect matchings and using arc diagrams.

Findings

Reduces search space compared to previous methods

Provides an exhaustive generation of Skolem sequences

Translates the problem into perfect matchings with arc diagrams

Abstract

A Skolem sequence is a linear arrangement of the multiset, {1, 1, 2, 2, ..., n, n} such that if r in [n] appears in positions i and j, then |i-j| = r. We first translate the problem to a particular set of perfect matchings, then apply the method of generating trees for open arc diagrams to generate exhaustively all Skolem sequences of a given size. Tracking the arc length between pairs of vertices in an arc annotated diagram is the central task. Although we do not surpass previously known enumerative results, this method drastically reduces the search space compared to previously known methods.