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Avoiding abelian powers cyclically
Advances in Applied Mathematics ( IF 1.0 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.aam.2020.102095
Jarkko Peltomäki , Markus A. Whiteland

Abstract 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 ≥ | w | . Let 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 A ∞ ( 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 ≤ A ( 2 ) ≤ 8 , 3 ≤ A ( 3 ) ≤ 4 , 2 ≤ A ( 4 ) ≤ 3 , and A ( k ) = 2 for k ≥ 5 . Moreover, we show that A ∞ ( 2 ) = 4 , A ∞ ( 3 ) = 3 , and A ∞ ( 4 ) = 2 .

中文翻译:

周期性地避免阿贝尔幂

摘要 我们研究了阿贝尔幂的循环回避的新概念。如果对于在无限词 w ω 中出现的周期 m 的每个阿贝尔 N 次幂,我们有 m ≥ | ,则有限词 w 循环避免了阿贝尔 N 次幂。| | . 设 A ( k ) 是最小整数 N,使得对于所有 n,在 k 字母字母表上存在一个长度为 n 的单词,它可以循环避免阿贝尔 N 次幂。设 A ∞ ( k ) 是最小整数 N,使得在 k 字母字母表上存在任意长的单词,可以循环避免阿贝尔 N 次幂。我们证明5≤A(2)≤8,3≤A(3)≤4,2≤A(4)≤3,对于k≥5,A(k)=2。此外,我们证明 A ∞ ( 2 ) = 4 、A ∞ ( 3 ) = 3 和 A ∞ ( 4 ) = 2 。
更新日期:2020-10-01
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