Counting occurrences of a pattern in a binary word

TL;DR

This paper introduces a new method called 'lexicographic extreme referencing' to compute the number of occurrences of a pattern in binary words, providing formulas and algorithms for counting and constructing such words.

Contribution

It presents a novel referencing method and algorithms for counting pattern occurrences and constructing primitive words in binary sequences.

Findings

Derived a formula for pattern occurrences in binary words.

Developed algorithms for constructing words with specific pattern counts.

Discussed approaches for counting words with exact pattern repetitions.

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 $B_{n,p}(k)$, 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 $B_{n,p}(k)$ via primitive words.