论文标题
无方向的重复阈值和无方向的模式避免
The undirected repetition threshold and undirected pattern avoidance
论文作者
论文摘要
对于有理数的$ r $，使$ 1 <r \ leq 2 $，一个无方向的$ r $ - 功率是$ xyx'$的单词，其中$ x $单词是非expprenty的，$ x'$ in $ \ {x，x，x，x^r \} $，我们有$ | xyx'| xyx'|/| xy | xy | xy | = r $。 $ k $字母的无向重复阈值，表示为$ \ mbox {urt}（k）$，是所有$ r $的集合中的最低限额，因此在$ k $ netters上可以避免无方向的$ r $ - 力量。我们首先证明$ \ mbox {urt}（3）= \ tfrac {7} {4} $。然后，我们证明$ \ mbox {urt}（k）\ geq \ tfrac {k-1} {k-2} $ for All $ k \ geq 4 $。我们推测所有$ k \ geq 4 $ $ \ mbox {urt}（k）= \ tfrac {k-1} {k-2} $ for All $ k \ geq 4 $，我们确认了$ k \ in \ in \ in \ in \ in \ {4,5，\ ldots，21 \}的猜想。特别是，我们发现每种二进制模式的无方向性避免索引。这是单词2019年呈现的论文的扩展版本，它包含了新的和改进的结果。
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.
