Relations on words

Rigo, Michel

doi:10.1016/j.indag.2016.11.018

Cited by 15 publications

(8 citation statements)

References 91 publications

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“…Many classical questions in combinatorics on words can be considered in this binomial context [27,29]. Avoiding binomial squares and cubes is considered in [26].…”

Section: Basicsmentioning

confidence: 99%

Computing the k-binomial Complexity of the Thue–Morse Word

Lejeune

Leroy

Rigo

2019

Developments in Language Theory

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Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this kbinomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue-Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue-Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form Ψ ℓ (w) for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue-Morse word by introducing a relevant new equivalence relation.

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“…Many classical questions in combinatorics on words can be considered in this binomial context [27,29]. Avoiding binomial squares and cubes is considered in [26].…”

Section: Basicsmentioning

confidence: 99%

Computing the k-binomial Complexity of the Thue–Morse Word

Lejeune

Leroy

Rigo

2019

Developments in Language Theory

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“…However, to highlight particular combinatorial properties of the infinite word of interest, other complexity measures such as abelian, k-abelian, cyclic, privileged, and k-binomial complexities have been introduced. See, for instance, [18,9,3,16,19]. In most cases, one considers the quotient of the language L(x) by a convenient equivalence relation ∼ and the corresponding complexity function therefore maps n ∈ N to #(L n (x)/∼).…”

Section: Introductionmentioning

confidence: 99%

Characterizations of families of morphisms and words via binomial complexities

Rigo¹,

Stipulanti²,

Whiteland³

2022

Preprint

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Two words are k-binomially equivalent, if each word of length at most k occurs as a subword, or scattered factor, the same number of times in both words. The k-binomial complexity of an infinite word maps the natural n to the number of k-binomial equivalence classes represented by its factors of length n. Inspired by questions raised by Lejeune, we study the relationships between the k and (k + 1)-binomial complexities; as well as the link with the usual factor complexity. We show that pure morphic words obtained by iterating a Parikh-collinear morphism, i.e. a morphism mapping all words to words with bounded abelian complexity, have bounded k-binomial complexity. In particular, we study the properties of the image of a Sturmian word by an iterate of the Thue-Morse morphism.

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“…Thus one can regard the notion of k-binomial equivalence as gradually bridging the gap between abelian equivalence (k = 1) and equality (k = +∞). An independent generalization of abelian equivalence is k-abelian equivalence where one counts factors of length at most k [9]; for more details, see [24].…”

Section: Introductionmentioning

confidence: 99%

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

Lü¹,

Chen²,

Zhou³

et al. 2021

Preprint

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In this paper, we study the 2-binomial complexity b tm,2 (n) of the generalized Thue-Morse words t m for every integer m ≥ 3. We obtain the exact value of b tm,2 (n) for every integer n ≥ m 2 . As a consequence, b tm,2 (n) is ultimately periodic with period m 2 . This result partially answers a question of M. Lejeune, J. Leroy and M. Rigo [Computing the k-binomial complexity of the Thue-Morse word, J. Comb. Theory Ser. A, 176 (2020) 105284].

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Relations on words

Cited by 15 publications

References 91 publications

Computing the k-binomial Complexity of the Thue–Morse Word

Computing the k-binomial Complexity of the Thue–Morse Word

Characterizations of families of morphisms and words via binomial complexities

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

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