Equations Enforcing Repetitions Under Permutations

The notion of repetition of factors in words is central to combinatorics on words. A recent generalisation of this concept considers repetitions under permutations: give an alphabet \(\Sigma \) and a morphism or antimorphism f on \(\Sigma ^*\), whose restriction to \(\Sigma \) is a permutation, w is an [f]-repetition if there exists \(\gamma \in \Sigma ^*\) such that \(w=f^{i_1}(\gamma )f^{i_2}(\gamma )\cdots f^{i_k}(\gamma )\), for some \(k\ge 2\). 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 [f]-repetitions.