Combinatorial Properties of primitive words with Non-primitive Product

TL;DR

This paper characterizes primitive words over an alphabet where the concatenation of two such words results in a non-primitive word, providing formulas for counting such pairs and analyzing their asymptotic behavior.

Contribution

It offers a complete description of primitive word pairs with non-primitive product and derives combinatorial formulas for their enumeration.

Findings

Derived a complete characterization of primitive word pairs with non-primitive concatenation.

Provided a combinatorial formula for counting such pairs based on word length and alphabet size.

Analyzed the asymptotic behavior of the count as word length and alphabet size grow large.

Abstract

Let $\mathcal{A}$ be an alphabet of size $n\ge 2$. In this paper, we give a complete description of primitive words $p\neq q$ over an alphabet $\mathcal{A}$ of size $n\geq2$ such that $pq$ is non-primitive and $|p|=2|q|$. In particular, if $l$ is s a positive integer, we count the cardinality of the set $\mathcal{E}(l,\mathcal{A})$ of all couples $(p,q)$ of primitive words such that $|p|=2|q|=2l$ and $pq$ is non-primitive. Then we give a combinatorial formula for this cardinality and its asymptotic behavior, as $l$ or $n$ goes to infinity.