Patterns can be generalized by adding functional dependencies between variables. Here, we consider involutions ϑ, that is, ϑ(ϑ(*a*)) = *a* for all letters *a*, and their morphic/antimorphic extensions on words, that is, ϑ(*uv*) = ϑ(*u*)ϑ(*v*) for the morphic and ϑ(*uv*) = ϑ(*v*)ϑ(*u*) for the antimorphic case. A pattern *p* in this setting consists now of (word) variables and function variables and it is avoided in some infinite word **w** if there exists no substitution of (word) variables by words and function variables by involutions for *p* such that the result is a factor of **w**. The morphic and antimorphic cases are usually considered separately.

The avoidance indices of all unary patterns under involution are known. See

B. Bischoff, J. Currie, D. Nowotka, *Unary Patterns With Involution*, (reference to be completed).

Let *p* be a pattern consisting of only one (word) variable and at most one function variable. Then for both the morphic and antimorphic case *p* is

- avoidable over three letters, if
*p*is of length 3 and not in {ααα, ϑ(α)ϑ(α)ϑ(α)}, - unavoidable, if
*p*is in {α, ϑ(α), αϑ(α), ϑ(α)α}, and - avoidable over two letters otherwise.

-- DirkNowotka - 12 Mar 2012

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Topic revision: r1 - 2012-03-12 - DirkNowotka

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