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This is a Quantitative Study study.

October 20, 2025Open Access

Counting occurrences of a pattern in a binary word

Key Points

The study provides a formula to compute occurrences of a binary word in another, enhancing pattern recognition.
Using lexicographic extreme referencing, the study constructs binary words that contain a specific pattern exactly k times.
An algorithm for generating primitive binary words with respect to a pattern is also developed, focusing on their combinatorial properties.
This exploration into binary patterns offers novel insights into generating and counting word occurrences.

Abstract

Enumerating the number of times one word occurs in another is a much-studied combinatorial subject. By utilizing a method that we call ``lexicographic extreme referencing'', we provide a formula for computing occurrences of one binary word in another. We then study B₍, ₏ (k), the number of binary words of length n containing a given word p exactly k times. For this purpose, we first use lexicographic extreme referencing to provide an algorithm for constructing all words w that contain a given word p. Afterward, we give a modified version of this algorithm for constructing the subset of binary words that are ``primitive'' with respect to p, and we discuss approaches for finding B₍, ₏ (k) via primitive words.

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Cite This Study

Rong Tian (Thu,) studied this question.

synapsesocial.com/papers/68f5fcdc8d54a28a75cf2468 https://doi.org/https://doi.org/10.48550/arxiv.2507.03205