On the 2-binomial complexity of the generalized Thue-Morse words

Lü, Xiaotao; Chen, Jin; Zhou, Wen; Wen, Wu

doi:10.48550/arxiv.2112.05347

Cited by 1 publication

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References 22 publications

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“…Boundary sets play an important role in the study of so-called k-abelian and k-binomial complexities of infinite words (for definitions, see [46]). For instance, computing the 2-binomial complexity of generalized Thue-Morse words [31] requires inspecting pairs of prefixes and suffixes of factors, which is again related to the boundary sequence when these prefixes and suffixes have equal length. The k-binomial complexities of images of binary words under powers of the Thue-Morse morphism are studied in [49]; there some general properties of boundary sequences of binary words are required (see [49,Lem.…”

Section: Motivation and Related Workmentioning

confidence: 99%

On Extended Boundary Sequences of Morphic and Sturmian Words

Rigo,

Stipulanti,

Whiteland

2024

Electron. J. Combin.

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Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for all pairs of factors of length $\ell$ separated by $n-\ell$ symbols. For the large class of addable abstract numeration systems $S$, we show that if an infinite word is $S$-automatic, then the same holds for its $\ell$-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and $S$-automatic words for which the boundary sequence is nevertheless $S$-automatic and conversely, $S$-automatic words with a boundary sequence that is not $S$-automatic. In the second part of the paper, we study the $\ell$-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.

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