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arXiv · 2202.09091

Combinatorial Properties of primitive words with Non-primitive Product

Othman Echi·Adel Khalfallah·Dhaker Kroumi
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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.

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Othman Echi, Adel Khalfallah, Dhaker Kroumi. 2022-02-18. Combinatorial Properties of primitive words with Non-primitive Product. https://arxiv.org/abs/2202.09091

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