Word Chain Generators for Prefix Normal Words

TL;DR

This paper explores properties of prefix normal words, introduces new methods for relating words of the same length through word chains and generators, and addresses open problems like enumeration and testing.

Contribution

It presents new insights into prefix normal words and introduces novel word chain and generator techniques for analyzing their properties.

Findings

Characterizes factors responsible for non-prefix normality

Introduces word chains and generators as tools for relating words

Provides new approaches to enumeration and testing of prefix normal words

Abstract

In 2011, Fici and Lipt\'ak introduced prefix normal words. A binary word is prefix normal if it has no factor (substring) that contains more occurrences of the letter 1 than the prefix of the same length. Among the open problems regarding this topic are the enumeration of prefix normal words and efficient testing methods. We show a range of characteristics of prefix normal words. These include properties of factors that are responsible for a word not being prefix normal. With word chains and generators, we introduce new ways of relating words of the same length to each other.