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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.12215 |
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| _version_ | 1866910213036048384 |
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| author | Li, Shuo Song, Yuan |
| author_facet | Li, Shuo Song, Yuan |
| contents | A \emph{square} is a word of the form $uu$, where $u$ is a nonempty finite word. Given a finite word $w$ of length $n$, let $[w]$ denote the corresponding \emph{circular word}, i.e., the set of all cyclic rotations of $w$. We study the number of distinct square factors of the elements of $[w]$. Amit and Gawrychowski first showed that this number is upper bounded by $3.14n$. In a recent article, Charalampopoulos et al. improved this upper bound to $1.8n$ and conjectured that the sharp upper bound is $1.5n$. In this note, we improve this upper bound to $\frac{5}{3}n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12215 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Tighter Upper Bound for the Number of Distinct Squares in Circular Words Li, Shuo Song, Yuan Combinatorics A \emph{square} is a word of the form $uu$, where $u$ is a nonempty finite word. Given a finite word $w$ of length $n$, let $[w]$ denote the corresponding \emph{circular word}, i.e., the set of all cyclic rotations of $w$. We study the number of distinct square factors of the elements of $[w]$. Amit and Gawrychowski first showed that this number is upper bounded by $3.14n$. In a recent article, Charalampopoulos et al. improved this upper bound to $1.8n$ and conjectured that the sharp upper bound is $1.5n$. In this note, we improve this upper bound to $\frac{5}{3}n$. |
| title | A Tighter Upper Bound for the Number of Distinct Squares in Circular Words |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.12215 |