2021/07/15 by Patawar, M., Kapoor, K.
#68R15 #Combinatorics (math.CO) #F.2.2 #FOS: Mathematics #G.2.1
paper · doi:10.48550/arxiv.2107.07473
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 (n)/(7). We show that the squares in the longest sequence of consecutive FS-double squares are conjugates.