Another generalization of abelian equivalence: Binomial complexity of infinite words

Rigo, Michel; Salimov, Pavel

doi:10.1016/j.tcs.2015.07.025

Cited by 54 publications

(56 citation statements)

References 16 publications

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“…Another motivation, close to combinatorics on words, stems from the study of k-binomial equivalence of finite words and k-binomial complexity of infinite words (see [23] for more details). Given two words of the same length, they are k-binomially equivalent if they have the same multiset of scattered factors of length k, also known as k-spectrum ( [1,18,24]).…”

Section: Introductionmentioning

confidence: 99%

Reconstructing Words from Right-Bounded-Block Words

Fleischmann

Lejeune

Manea

et al. 2020

Lecture Notes in Computer Science

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A reconstruction problem of words from scattered factors asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w ∈ {a, b} * can be reconstructed from the number of occurrences of at most min(|w|a, |w|b) + 1 scattered factors of the form a i b, where |w|a is the number of occurrences of the letter a in w. Moreover, we generalize the result to alphabets of the form {1,. .. , q} by showing that at most q−1 i=1 |w|i (q − i + 1) scattered factors suffices to reconstruct w. Both results improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here. M. Lejeune-Supported by a FNRS fellowship. F. Manea-Supported by the DFG grant MA 5725/2-1.

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Section: Introductionmentioning

confidence: 99%

Reconstructing Words from Right-Bounded-Block Words

Fleischmann

Lejeune

Manea

et al. 2020

Lecture Notes in Computer Science

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“…We collect the known facts about the k-binomial complexity. For all k ≥ 2, Sturmian words have a k-binomial complexity which is the same as their factor complexity, i.e., b x,k (n) = n+1 for all n. Since b x,k (n) ≤ b x,k+1 (n), the proof consists in showing that any two distinct factors of length n occurring in a given Sturmian word are never 2-binomially equivalent [17,Thm. 7].…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 3. [17] Let k ≥ 1. There exists C k > 0 such that the k-binomial complexity of the Thue-Morse word t satisfies b t,k (n) ≤ C k for all n ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

Templates for the k-binomial complexity of the Tribonacci word

Lejeune

Rigo

Rosenfeld

2020

Advances in Applied Mathematics

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Consider the k-binomial equivalence: two finite words are equivalent if they share the same subwords of length at most k with the same multiplicities. With this relation, the k-binomial complexity of an infinite word x maps the integer n to the number of pairwise nonequivalent factors of length n occurring in x. In this paper based on the notion of template introduced by Currie et al., we show that, for all k ≥ 2, the k-binomial complexity of the Tribonacci word coincides with its usual factor complexity p(n) = 2n + 1. A similar result was already known for Sturmian words but the proof relies on completely different techniques that seemingly could not be applied for Tribonacci.

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“…Two words are called abelian equivalent if they contain the same number of occurrences of each letter, or, equivalently, if they are permutations of each other. In the recent years there is a growing interest in abelian properties of words, as well as modifications of the notion of abelian equivalence [1, 6,22,26,29]. One such modification is the notion of k-abelian equivalence: two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. For k = 1, the notion of k-abelian equivalence coincides with the notion of abelian equivalence, and when k is greater than half of the length of the words, k-abelian equivalence means equality.…”

Section: Introductionmentioning

confidence: 99%

On k-abelian palindromes

Cassaigne

Karhumäki

Puzynina

2018

Information and Computation

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A word is called a palindrome if it is equal to its reversal. In the paper we consider a k-abelian modification of this notion. Two words are called k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. We say that a word is a k-abelian palindrome if it is k-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length n can contain at most n + 1 distinct palindromes as its factors; such words are called rich. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called poor. It is easy to see that there are no 1-abelian poor words, and there exist words containing Θ(n 2) distinct 1-abelian palindromes. We analyze these notions with respect to k-abelian equivalence. Our main results concern poor words: We show that in the k-abelian case there exist infinite words containing finitely many distinct k-abelian palindromic factors. We also make some observation concerning rich words, namely, we show that there exist finite rich words containing Θ(n 2) distinct k-abelian palindromes as their factors. Therefore, for poor words the situation resembles that of usual palindromes, while for rich words it is similar to the 1-abelian case.

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Another generalization of abelian equivalence: Binomial complexity of infinite words

Cited by 54 publications

References 16 publications

Reconstructing Words from Right-Bounded-Block Words

Reconstructing Words from Right-Bounded-Block Words

Templates for the k-binomial complexity of the Tribonacci word

On k-abelian palindromes

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