Equations enforcing repetitions under permutations

© 2020 Elsevier B.V. The notion of repetition of factors in words is central to combinatorics on words. A recent generalisation of this concept considers repetitions under permutations: given an alphabet Σ and a morphism or antimorphism f on Σ∗, whose restriction to Σ is a permutation, w is an [f]-repetition if there exists γ∈Σ∗, an integer k≥2, and the positive integers i1,…,ik such that w=fi1(γ)fi2(γ)⋯fik(γ). 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 [f]-repetitions.