The undirected repetition threshold and undirected pattern avoidance

06/12/2020

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by   James D. Currie, et al.

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For a rational number r such that 1<r≤ 2, an undirected r-power is a word of the form xyx', where the word x is nonempty, the word x' is in {x,x^R}, and we have |xyx'|/|xy|=r. The undirected repetition threshold for k letters, denoted (k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that (3)=74. Then we show that (k)≥k-1k-2 for all k≥ 4. We conjecture that (k)=k-1k-2 for all k≥ 4, and we confirm this conjecture for k∈{4,5,…,21}. We then consider related problems in pattern avoidance; in particular, we find the undirected avoidability index of every binary pattern. This is an extended version of a paper presented at WORDS 2019, and it contains new and improved results.