Existence of an infinite ternary 64-abelian square-free word

Huova, Mari

doi:10.1051/ita/2014012

Cited by 4 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The paper also deals with bounds on enumeration results in that context of avoidance. About other avoidance results, also see [50,52,53] Remark 40. A variant of the notion of repetition is considered in [51], a word is a strongly ℓ-abelian nth power, if it is ℓ-abelian equivalent to a 'classical' nth power.…”

Section: K-abelian Equivalencementioning

confidence: 84%

Relations on words

Rigo

2017

Indagationes Mathematicae

View full text Add to dashboard Cite

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.In the second part, we mainly focus on abelian equivalence, k-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and M -equivalence. In particular, some new refinements of abelian equivalence are introduced.

show abstract

Section: K-abelian Equivalencementioning

confidence: 84%

Relations on words

Rigo

2017

Indagationes Mathematicae

View full text Add to dashboard Cite

show abstract

“…We know that for k = 3 there exist words of length 100000 avoiding 3-abelian squares. The avoidability in infinite words of k-abelian squares in a ternary alphabet is only known for large values of k (k ≥ 64) (see [11]).…”

Section: Bounded K-abelian Complexity and K-abelian Repetitionsmentioning

confidence: 99%

On a generalization of Abelian equivalence and complexity of infinite words

Karhumäki

Saarela

Zamboni

2013

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z + ∪ {+∞}. Let k ∈ Z + ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A * are said to be k-Abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations ∼ k on A * , bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +∞). We show that the number of k-Abelian equivalence classes of words of length n grows polynomially, although the degree is exponential in k. Given an infinite word ω ∈ A N , we consider the associated complexity function P (k) ω : N → N which counts the number of k-Abelian equivalence classes of factors of ω of length n. We show that the complexity function P (k) is intimately linked with periodicity. More precisely we define an auxiliary function q k : N → N and show that if P (k) ω (n) < q k (n) for some k ∈ Z + ∪ {+∞} and n ≥ 0, the ω is ultimately periodic. Moreover if ω is aperiodic, then P (k) ω (n) = q k (n) if and only if ω is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemerédi's theorem, we show that if ω has bounded k-Abelian complexity, then for every D ⊂ N with positive upper density and for every positive integer N, there exists a k-Abelian N power occurring in ω at some position j ∈ D.

show abstract

“…Please note that between the first presentation of this paper and its publication, the main questions of avoidability for k-abelian squares and cubes were fully settled [6,13,14]. Thus, it is now known that over a binary alphabet 2-abelian cubes are avoidable, while only a ternary alphabet is needed in order to avoid 3-abelian squares [14].…”

Section: Introductionmentioning

confidence: 99%