The Length of the Longest Sequence of Consecutive FS-double Squares in a word

TL;DR

This paper investigates the structure and maximum length of sequences of consecutive FS-double squares in words, establishing an upper bound of n/7 for such sequences and analyzing their properties.

Contribution

It introduces the concept of FS-double squares, characterizes their structure, and proves an upper bound on the length of consecutive sequences in words.

Findings

Maximum length of consecutive FS-double squares is at most n/7.

Squares in the longest sequence are conjugates.

Conditions for generating sequences of FS-double squares are identified.

Abstract

A square is a concatenation of two identical words, and a word $w$ is said to have a square $yy$ if $w$ can be written as $xyyz$ for some words $x$ and $z$. It is known that the ratio of the number of distinct squares in a word to its length is less than two and any location of a word could begin with at most two rightmost distinct squares. A square whose first location starts with the last occurrence of two distinct squares is an FS-double square. We explore and identify the conditions to generate a sequence of locations in a word that starts with FS-double squares. We first find the structure of the smallest word that begins with two consecutive FS-double squares and obtain its properties that enable to extend the sequence of FS-double squares. It is proved that the length of the longest sequence of consecutive FS-double squares in a word of length $n$ is at most $\frac{n}{7}$. We show that the squares in the longest sequence of consecutive FS-double squares are conjugates.