On a Greedy Algorithm to Construct Universal Cycles for Permutations

Abstract

A universal cycle for permutations of length [Formula: see text] is a cyclic word or permutation, any factor of which is order-isomorphic to exactly one permutation of length [Formula: see text], and containing all permutations of length [Formula: see text] as factors. It is well known that universal cycles for permutations of length [Formula: see text] 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 [Formula: see text], which is based on applying a greedy algorithm to a permutation of length [Formula: see text]. We prove that this approach gives a unique universal cycle [Formula: see text] for permutations, and we study properties of [Formula: see text].