Unary patterns under permutations

Thue characterized completely the avoidability of unary patterns. Adding function variables gives a general setting capturing avoidance of powers, avoidance of patterns with palindromes, avoidance of powers under coding, and other questions of recent interest. Unary patterns with permutations have been previously analysed only for lengths up to 3. Consider a pattern p=πi1(x)…πir(x), with r≥4, x a word variable over an alphabet Σ and πij function variables, to be replaced by morphic or antimorphic permutations of Σ. If |Σ|≥3, we show the existence of an infinite word avoiding all pattern instances having |x|≥2. If |Σ|=3 and all πij are powers of a single morphic or antimorphic π, the length restriction is removed. For the case when π is morphic, the length dependency can be removed also for |Σ|=4, but not for |Σ|=5, as the pattern xπ2(x)π56(x)π33(x) becomes unavoidable. Thus, in general, the restriction on x cannot be removed, even for powers of morphic permutations. Moreover, we show that for every positive integer n there exists N and a pattern πi1(x)…πin(x) which is unavoidable over all alphabets Σ with at least N letters and π morphic or antimorphic permutation.