On a Greedy Algorithm to Construct Universal Cycles for Permutations

2017/11/29 by Gao, Alice L. L., Kitaev, Sergey, Steiner, Wolfgang +1
#05A05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.10820

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.

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