Universal cycles for permutations

TL;DR

This paper constructs a universal cycle for permutations using only n+1 distinct integers, confirming a conjecture and advancing combinatorial cycle theory.

Contribution

It provides a construction of universal cycles for permutations with minimal distinct integers, proving a conjecture by Chung, Diaconis, and Graham.

Findings

Universal cycle for permutations constructed with n+1 integers

Proves the conjecture of Chung, Diaconis, and Graham

Optimal minimal number of integers used in the cycle

Abstract

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.