Unary Patterns of Size Four with Morphic Permutations

Abstract: We investigate the avoidability of unary patterns of size of four with morphic permutations. More precisely, we show that, for the positive integers $i,j,k$, the sizes of the alphabets over which a pattern $x \pi ^ {i} (x) \pi^{j}(x) \pi^{k}(x)$ is avoidable are an interval of the integers (where $x$ is a word variable and $\pi$ is a function variable with values in the set of all morphic permutations of the respective alphabets). We also show how to compute a good approximation of this interval. This continues the work of [Manea et al., 2015], where a complete characterisation of the avoidability of cubic patterns with permutations was given.