109 research outputs found

    Generalized trapezoidal words

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    The factor complexity function Cw(n)C_w(n) of a finite or infinite word ww counts the number of distinct factors of ww of length nn for each n0n \ge 0. A finite word ww of length w|w| is said to be trapezoidal if the graph of its factor complexity Cw(n)C_w(n) as a function of nn (for 0nw0 \leq n \leq |w|) is that of a regular trapezoid (or possibly an isosceles triangle); that is, Cw(n)C_w(n) increases by 1 with each nn on some interval of length rr, then Cw(n)C_w(n) is constant on some interval of length ss, and finally Cw(n)C_w(n) decreases by 1 with each nn on an interval of the same length rr. Necessarily Cw(1)=2C_w(1)=2 (since there is one factor of length 00, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of R2\mathbb{R}^2 made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet A\mathcal{A}, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when A3|\mathcal{A}| \geq 3, but we can describe all those that are rich.Comment: Major revisio

    Directive words of episturmian words: equivalences and normalization

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    Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.Comment: 15 page

    On Quasiperiodic Morphisms

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    Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.Comment: 12 page

    Quasiperiodic Sturmian words and morphisms

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    AbstractWe characterize all quasiperiodic Sturmian words: A Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity

    Detecting Episodes with Harmonic Sequences for Fugue Analysis

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    International audienceFugues alternate between instances of the subject and of other patterns, such as the counter-subject, and modulatory sections called episodes. The episodes play an important role in the overall design of a fugue: detecting them may help the analysis of the fugue, in complement to a subject and a counter-subject detection. We propose an algorithm to retrieve episodes in the fugues of the first book of Bach's Well-Tempered Clavier, starting from a symbolic score which is already track-separated. The algorithm does not use any information on subject or counter-subject occurrences, but tries to detect partial harmonic sequences, that is similar pitch contour in at least two voices. For this, it uses a substitution function considering "quantized partially overlapping intervals" [Lemström and Laine, 98] and a strict length matching for all notes, except for the first and the last one. On half of the tested fugues, the algorithm has correct or good results, enabling to sketch the design of the fugue

    A Characterization of Infinite LSP Words

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    G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of SS-adicity. More precisely we provide a finite set of morphisms SS and an automaton A{\cal A} such that an infinite word is LSP if and only if it is SS-adic and all its directive words are recognizable by A{\cal A}

    Powers in a class of A-strict standard episturmian words

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    This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the kk-bonacci word: a generalization of the Fibonacci word to a kk-letter alphabet (k2k\geq2).Comment: 26 pages; extended version of a paper presented at the 5th International Conference on Words, Montreal, Canada, September 13-17, 200

    Rhythm extraction from polyphonic symbolic music

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    International audienceWe focus on the rhythmic component of symbolic music similarity, proposing several ways to extract a monophonic rhythmic signature from a symbolic poly- phonic score. To go beyond the simple extraction of all time intervals between onsets (noteson extraction), we select notes according to their length (short and long extractions) or their intensities (intensity+/− extractions). Once the rhythm is extracted, we use dynamic programming to compare several sequences. We report results of analysis on the size of rhythm patterns that are specific to a unique piece, as well as experiments on similarity queries (ragtime music and Bach chorale variations). These results show that long and intensity+ extractions are often good choices for rhythm extraction. Our conclusions are that, even from polyphonic symbolic music, rhythm alone can be enough to identify a piece or to perform pertinent music similarity queries, especially when using wise rhythm extractions

    Quasiperiodic and Lyndon episturmian words

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    Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their "directive words". Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular "normalized" directive words, which we introduced in an earlier paper. Also of importance is the use of "return words" to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.Comment: 33 pages; minor change
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