Abstract
Let𝒜be an alphabet of sizen≥ 2. Our goal in this paper is to give a complete description of primitive wordsp≠qover𝒜such thatpqis non-primitive. As an application, we will count the cardinality of the setℰ(l,𝒜) of all couples (p,q) of distinct primitive words such that |p| = |q| =landpqis non-primitive, wherelis a positive integer. Then we give a combinatorial formula for the cardinalityε(n,l) of this set. The density in {(p,q) :p,qare distinct primitive words and |p| = |q| =l} of the setℰ(l,𝒜) is also discussed.
Highlights
Combinatorics on words is an area of research focusing on combinatorial properties of words applied to formal languages
The empty word over A will be denoted by ε, A∗ is the set of all words over A and A+ is the set of all nonempty words over A
In [18], Reis-Shyr have proved that every non-empty word which is not a power of a letter is a product of two primitive words
Summary
Combinatorics on words is an area of research focusing on combinatorial properties of words applied to formal languages It plays an important role in several mathematical research areas as well as theoretical computer science (see [2, 6, 9, 21]). Whether Q(A) is a context-free language or not is a well-known long-standing open problem posed by Domsi et al [4, 8] This problem was the origin of most of the combinatorial studies of primitive words. In [18], Reis-Shyr have proved that every non-empty word which is not a power of a letter is a product of two primitive words. In [20] Shyr-Yu gave a necessary condition when the product pqm is a k-power of a primitive word (i.e., pqm ∈ Q(k)(A)), with m, k ≥ 2.
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