On a Greedy Algorithm to Construct Universal Cycles for Permutations

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 Π n for permutations, and we study properties of Π n .

Shortened universal cycles for permutations

“…It would be interesting to investigate the smallest number of symbols needed to create a shortened universal cycle of a given length. As suggested in [7], it would also be interesting to determine a greedy algorithm for constructing shortened universal cycles for S n .…”