On gapped repeats in a cyclic Fibonacci word

2026-06-01

AI Summary

This study investigates the structure and enumeration of index pairs $(i, k)$ corresponding to all repeated substrings of length $s$ in circular Fibonacci words. By leveraging combinatorial mathematics, formal language theory, and intrinsic structural properties of Fibonacci words, the work provides the first complete characterization of occurrence patterns for such equal-length repeats—including gapped repetitions—and derives exact counting formulas. This result fills a critical gap in the systematic analysis of repeated substrings within circular Fibonacci words, establishing a theoretical foundation for further research in string algorithms and combinatorial structures on words.

Abstract

In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(ι,κ)$ of indices such that the word of length $s$ from $ι$ is equal to the word of length $s$ from $κ$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.

Problem

• gapped repeats
• cyclic Fibonacci word
• equal substrings
• index pairs

Innovation

• cyclic Fibonacci word
• gapped repeats
• combinatorics on words
• index pairs
• substring repetitions