The undirected repetition threshold and undirected pattern avoidance

Abstract

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 URT(k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that URT(3) = 7 4 . Then we show that URT(k) ≥ k−1 k−2 for all k ≥ 4. We conjecture that URT(k) = k−1 k−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. MSC 2010: 68R15

A family of formulas with reversal of arbitrarily high avoidability index

“…For example, avoiding xyxy and xyx R y simultaneously is equivalent to avoid xyx U y. Notice that the notion of undirected avoidability recently considered by Currie and Mol [1] corresponds to the case where every occurrence of every variable of the pattern/formula is equipped with − U .…”