Equations Enforcing Repetitions Under Permutations

Abstract

The notion of repetition of factors in words is central to combinatorics on words. A recent generalization of this concept considers repetitions under permutations: given an alphabet Σ and a morphism or antimorphism on Σ * , whose restriction to Σ is a permutation, is an [ ]-repetition if there exists ∈ Σ * , an integer ≥ 2, and the positive integers 1 , … , such that = 1 ( ) 2 ( ) ⋯ ( ). In this paper, we extend a series of classical repetition enforcing word equations to this general setting to obtain a series of word equations whose solutions are [ ]-repetitions.

Pseudo-solutions of word equations

“…The minimality of M now implies Z = M , hence Y = C. Example 10. In [4], the following claim is proved (see [4], Eq.1 and Theorem 21):…”

“…Of course, this may depend on the anticongruence in question. Some answers for particular equations and morphic permutations were given in [9,4]. The following theorem gives a general answer which makes it unnecessary to produce new results of this kind.…”

“…Third, and most importantly, the claim is a direct consequence of Theorem 8, whose proof should be compared to the rough sketch of the proof in [4]. Of course, this does not mean that we have found a short self-contained proof of the above claim.…”

“…The trivial instance of an anticongruence is equality. A more interesting example, which motivated our research, is given by morphic permutations introduced and studied in [4]. A morphic permutation is a permutation on Σ extended to a (length preserving) morphism on Σ * .…”

“…In the case of morphic permutations, it turned out that several classical results hold even in the generalized setting, while much less can be said about the antimorphic case. In [4], the authors announced, as the main result, a solution of the Lyndon and Schützenberger equation for morphic permutations. The proof is only sketched and is very involved.…”