Unary Patterns of Size Four with Morphic Permutations
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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 π ^ i (x) π^j(x) π^k(x) is avoidable are an interval of the integers (where x is a word variable and π 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.
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