17 research outputs found

    Abelian returns in Sturmian words

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    In this paper we study an abelian version of the notion of return word. Our main result is a new characterization of Sturmian words via abelian returns. Namely, we prove that a word is Sturmian if and only if each of its factors has two or three abelian returns. In addition, we describe the structure of abelian returns in Sturmian words, and discuss connections between abelian returns and periodicity

    Enumerating Abelian Returns to Prefixes of Sturmian Words

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    We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set APRu\mathcal{APR}_u of abelian returns of all prefixes of a Sturmian word uu in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set APRu\mathcal{APR}_u and we determine it for the characteristic Sturmian words.Comment: 19page

    Some properties of abelian return words (long abstract)

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    We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite

    All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words

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    We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f: ? ? ?, we construct an infinite binary word whose abelian exponents have limit superior growth rate f. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word

    On a generalization of Abelian equivalence and complexity of infinite words

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    In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and AA be a finite non-empty set. Two finite words uu and vv in AA^* are said to be kk-Abelian equivalent if for all xAx\in A^* of length less than or equal to k,k, the number of occurrences of xx in uu is equal to the number of occurrences of xx in v.v. This defines a family of equivalence relations k\thicksim_k on A,A^*, bridging the gap between the usual notion of Abelian equivalence (when k=1k=1) and equality (when k=+).k=+\infty). We show that the number of kk-Abelian equivalence classes of words of length nn grows polynomially, although the degree is exponential in k.k. Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of kk-Abelian equivalence classes of factors of ω\omega of length n.n. We show that the complexity function P(k)\mathcal {P}^{(k)} is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if Pω(k)(n)<qk(n)\mathcal {P}^{(k)}_{\omega}(n)<q^k(n) for some k \in \ints ^+ \cup {+\infty} and n0,n\geq 0, the ω\omega is ultimately periodic. Moreover if ω\omega is aperiodic, then Pω(k)(n)=qk(n)\mathcal {P}^{(k)}_{\omega}(n)=q^k(n) if and only if ω\omega is Sturmian. We also study kk-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if ω\omega has bounded kk-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer N,N, there exists a kk-Abelian NN power occurring in ω\omega at some position $j\in D.

    Relations on words

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    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, kk-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and MM-equivalence. In particular, some new refinements of abelian equivalence are introduced

    45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

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    We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f ⁣:NRf\colon \N \to \R, we construct an infinite binary word whose abelian exponents have limit superior growth rate ff. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word.</p
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