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

Document Type : Original paper

Authors

1 Department of Computer Science and Engineering, Indian Institute of Technology Guwahati

2 Department of Mathematics, Indian Institute of Technology Guwahati

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 two distinct squares which are appearing in the word for the last time. 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 under which a sequence of locations in a word starts with FS-double squares. We first find the structure of a word that begins with two consecutive FS-double squares and obtain its properties that enable us 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.

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References

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Communications in Combinatorics and Optimization
Volume 9, Issue 2
June 2024
Pages 263-277

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