703 research outputs found

    Constructions of words rich in palindromes and pseudopalindromes

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    A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called HH-rich words) is demonstrated. The correspondence between rich and HH-rich words is based on the operation SS acting over words over the alphabet {0,1}\{0,1\} and defined by S(u0u1u2)=v1v2v3S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots, where vi=ui1+uimod2v_i= u_{i-1} + u_i \mod 2. The operation SS enables us to construct a new class of rich words and a new class of HH-rich words. Finally, the operation SS is considered on the multiliteral alphabet Zm\mathbb{Z}_m as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and HH-rich words are obtained by application of SS on the generalized Thue--Morse words over the alphabet Z4\mathbb{Z}_4.Comment: 26 page

    Factor frequencies in generalized Thue-Morse words

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    We describe factor frequencies of the generalized Thue-Morse word t_{b,m} defined for integers b greater than 1, m greater than 0 as the fixed point starting in 0 of the morphism \phi_{b,m} given by \phi_{b,m}(k)=k(k+1)...(k+b-1), where k = 0,1,..., m-1 and where the letters are expressed modulo m. We use the result of A. Frid, On the frequency of factors in a D0L word, Journal of Automata, Languages and Combinatorics 3 (1998), 29-41 and the study of generalized Thue-Morse words by S. Starosta, Generalized Thue-Morse words and palindromic richness, arXiv:1104.2476v2 [math.CO].Comment: 11 page

    On the critical exponent of generalized Thue-Morse words

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    For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

    On the critical exponent of generalized Thue-Morse words

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    Automata, Logic and Semantic

    On winning shifts of marked uniform substitutions

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    The second author introduced with I. T\"orm\"a a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence α1\alpha_1, α2\alpha_2, \ldots of integers such that on round nn the player AA chooses a subset SnS_n of size αn\alpha_n of some fixed finite alphabet and the player BB picks a letter from the set SnS_n. The outcome is determined by whether the word obtained by concatenating the letters BB picked lies in a prescribed target set XX (a win for player AA) or not (a win for player BB). Typically, we consider XX to be a subshift. The winning shift W(X)W(X) of a subshift XX is defined as the set of choice sequences for which AA has a winning strategy when the target set is the language of XX. The winning shift W(X)W(X) mirrors some properties of XX. For instance, W(X)W(X) and XX have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that W(X)W(X) and XX have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.Comment: Extended version of a paper presented at RuFiDiM I

    Generalized Thue-Morse words and palindromic richness

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    We prove that the generalized Thue-Morse word tb,m\mathbf{t}_{b,m} defined for b2b \geq 2 and m1m \geq 1 as tb,m=(sb(n)modm)n=0+\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}, where sb(n)s_b(n) denotes the sum of digits in the base-bb representation of the integer nn, has its language closed under all elements of a group DmD_m isomorphic to the dihedral group of order 2m2m consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms ΘDm\Theta \in D_m, we show that tb,m\mathbf{t}_{b,m} is saturated by Θ\Theta-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that tb,m\mathbf{t}_{b,m} is DmD_m-rich. We also calculate the factor complexity of tb,m\mathbf{t}_{b,m}.Comment: 11 page
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