无向重复阈值与无向模式回避
The undirected repetition threshold and undirected pattern avoidance
摘要：
对于一个有理数$r$，使得$1<r\leq 2$，一个无向$r$ -幂是一个形式为$xyx'$的字，其中字$x$是非空的，字$x'$在$\{x,x^R\}$中，我们有$|xyx'|/|xy|=r$。用于$k$字母的无向重复阈值（表示为$\mbox{URT}(k)$）是所有$r$的集合的下确界，使得在$k$字母上可避免无向$r$ -幂。我们首先证明了$\mbox{URT}(3)=\tfrac{7}{4}$ .然后我们证明了对于所有的$k\geq 4$，$\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$。我们猜想，$\mbox{URT}(k)=\tfrac{k-1}{k-2}$的所有$k\geq 4$，我们证实了这一猜想为$k\in\{4,5,\ldots,21\}.$，我们然后考虑相关的问题，在模式回避，特别是，我们发现的无向回避指数的每一个二进制模式。这是在WORDS 2019上发表的一篇论文的扩展版本，它包含了新的和改进的结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
