On a Greedy Algorithm to Construct Universal Cycles for Permutations

TL;DR

This paper introduces a simple greedy algorithm to construct universal cycles for permutations, providing a new, straightforward method that guarantees a unique cycle and explores its properties.

Contribution

It presents a novel greedy algorithm-based approach for constructing universal cycles for permutations, simplifying previous complex methods.

Findings

The greedy algorithm produces a unique universal cycle for permutations.

The paper proves the correctness and uniqueness of the cycle generated.

Properties of the constructed universal cycle are analyzed.

Abstract

A universal cycle for permutations of length $n$ is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length $n$, and containing all permutations of length $n$ as factors. It is well known that universal cycles for permutations of length $n$ exist. However, all known ways to construct such cycles are rather complicated. For example, in the original paper establishing the existence of the universal cycles, constructing such a cycle involves finding an Eulerian cycle in a certain graph and then dealing with partially ordered sets. In this paper, we offer a simple way to generate a universal cycle for permutations of length $n$, which is based on applying a greedy algorithm to a permutation of length $n-1$. We prove that this approach gives a unique universal cycle $\Pi_n$ for permutations, and we study properties of $\Pi_n$.