The undirected repetition threshold and undirected pattern avoidance

TL;DR

This paper investigates the undirected repetition threshold for words over finite alphabets, establishing exact values for small cases, providing bounds for larger cases, and exploring pattern avoidance, including the undirected avoidability index of binary patterns.

Contribution

It determines the undirected repetition threshold for 3 letters, establishes a lower bound for larger alphabets, conjectures an exact value for all larger cases, and analyzes pattern avoidance including the avoidability index.

Findings

URT(3)=7/4

URT(k)≥(k-1)/(k-2) for k≥4

Confirmed the conjecture URT(k)=(k-1)/(k-2) for 4≤k≤21

Abstract

For a rational number $r$ such that $1<r\leq 2$, 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 $\mbox{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mbox{URT}(3)=\tfrac{7}{4}$. Then we show that $\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mbox{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in\{4,5,\ldots,21\}.$ 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.