Abelian repetitions in partial words

AbstractWe study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes. First, we look at the avoidance of abelian pth powers in infinite partial words, where p>2, extending recent results regarding the case where p=2. We investigate, for a given p, the smallest alphabet size needed to construct an infinite partial word with finitely or infinitely many holes that avoids abelian pth powers. We construct in particular an infinite binary partial word with infinitely many holes that avoids 6th powers. Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases. Finally, we prove that we cannot avoid abelian pth powers under arbitrary insertion of holes in an infinite word

Avoiding abelian powers cyclically

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 $w^\omega$, 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 $\mathcal{A}_\infty(4) = 2$.</p