论文标题
周期性地避免阿贝里亚人的力量
Avoiding abelian powers cyclically
论文作者
论文摘要
我们研究了循环避免阿伯利亚力量的新概念。一个有限的单词$ w $避免了Abelian $ n $ - 如果每个Abelian $ n $ n $ n $ n $ n $ n $ n $ - 在无限单词$ w^ω$中出现的时期$ m $，我们有$ m \ geq | w | $。令$ \ MATHCAL {a}（k）$为最小整数$ n $，使得对于所有$ n $，都存在一个长度$ n $的单词，$ k $ n $ - $ k $ - 字母，避免了Abelian $ n $ n $ n $ - 周期性的能力。令$ \ Mathcal {a} _ \ infty（k）$为最小整数$ n $，使得在$ k $ - letter字母上任意长的单词避免了abelian $ n $ n $ n $ cycly-cly-clycly-ncland of。 我们证明，$ 5 \ leq \ Mathcal {a}（2）\ leq 8 $，$ 3 \ leq \ Mathcal {a}（3）（3）\ leq 4 $，$ 2 \ leq \ leq \ Mathcal {a}（4）（4）\ leq 3 $，and $ \ \ \ \ \ \ \ \ \ Mathcal {a a}（a）此外，我们表明$ \ MATHCAL {A} _ \ infty（2）= 4 $，$ \ MATHCAL {a} _ \ infty（3）= 3 $和$ \ MATHCAL {a} _ \ infty（4）= 2 $。
We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^ω$, we have $m \geq |w|$. Let $\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that there exist arbitrarily long words over a $k$-letter alphabet that avoid abelian $N$-powers cyclically. We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$, $2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$. Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) = 3$, and $\mathcal{A}_\infty(4) = 2$.
