Exact Net-Occurrence Counts in Purely Morphic Regular Epistandard Words

2026-10-03

AI Summary

This study investigates whether the invariant occurrence of exactly three net occurrences in Fibonacci words constitutes an instance of a broader morphological phenomenon. To address this, we examine finite approximations of purely morphic standard Sturmian words over specific alphabets, employing palindromic prefixes, return word decompositions, and overlapping net occurrence covering techniques to establish rigorous proofs. The work reveals a strict dichotomy governed by exponent values: the number of net occurrences is three when the final exponent equals one, and two when it is at least two. Furthermore, it confirms that all non-initial approximations in the d-bonacci case contain exactly three net occurrences. By successfully embedding the Fibonacci phenomenon within a more general theoretical framework, this research deepens our understanding of the intrinsic structures underlying combinatorics on words.

Abstract

Finite Fibonacci words have recently been shown to contain exactly three net occurrences---occurrences of repeated factors whose one-letter left and right extensions are unique. This unexpectedly small constant raises a natural question: is it peculiar to the Fibonacci recurrence, or part of a broader morphic phenomenon? We study a family over the alphabet $\{0,1,\ldots,d-1\}$ determined by positive integers $e_0,\ldots,e_{d-1}$. For each letter $a$, let $L_a$ be the morphism that fixes $a$ and maps every other letter $b$ to $ab$; we consider the finite approximants $S_m=\mu^m(0)$ generated by $\mu=L_0^{e_0}\cdots L_{d-1}^{e_{d-1}}$. These words are the period-aligned finite approximants of the purely morphic regular epistandard family considered here. We prove a sharp dichotomy: for every $m\ge2$, $S_m$ has exactly three net occurrences when $e_{d-1}=1$, and exactly two when $e_{d-1}\ge2$; the initial approximant is also completely classified. The proof combines palindromic prefixes, return-word factorizations, and overlapping net-occurrence covers. As a consequence, when all exponents are equal to one---the standard $d$-bonacci case---every noninitial period-aligned finite approximant has exactly three net occurrences, placing the Fibonacci phenomenon in a wider epistandard framework.

Problem

• net occurrences
• Fibonacci words
• epistandard words
• purely morphic words
• combinatorics on words

Innovation

• net occurrences
• purely morphic words
• epistandard words
• return-word factorization
• palindromic prefixes