Containing all permutations

TL;DR

This paper surveys various questions about the shortest objects containing all permutations of a certain length, synthesizes existing answers, introduces new related questions, and improves an upper bound for one of these problems.

Contribution

It provides a comprehensive survey of permutation containment problems, introduces new related questions, and establishes an improved upper bound for a specific permutation containment problem.

Findings

Survey of historical and recent questions on permutation containment.

Introduction of infinitely many related questions.

Improved upper bound for a permutation containment problem.

Abstract

Numerous versions of the question "what is the shortest object containing all permutations of a given length?" have been asked over the past fifty years: by Karp (via Knuth) in 1972; by Chung, Diaconis, and Graham in 1992; by Ashlock and Tillotson in 1993; and by Arratia in 1999. The large variety of questions of this form, which have previously been considered in isolation, stands in stark contrast to the dearth of answers. We survey and synthesize these questions and their partial answers, introduce infinitely more related questions, and then establish an improved upper bound for one of these questions.