# Kalle Saari

The University of Winnipeg, Winnipeg, Manitoba, Canada

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## Publications (23)7.95 Total impact

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##### Article: Square-free words with square-free self-shuffles
James D. Currie · Kalle Saari ·
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ABSTRACT: We answer a question of Harju: For every n≥3 there is a square-free ternary word of length n with a square-free self-shuffle.
The electronic journal of combinatorics 01/2014; 21(1). · 0.49 Impact Factor
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##### Article: Suffix conjugates for a class of morphic subshifts
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ABSTRACT: Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X, T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is the shift map. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a in A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0} such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is primitive and f(A) is a suffix code, then there exists a mapping H: calS --> calS such that (calS, H) is a topological dynamical system and \pi: (calS, H) --> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In the special case when f is the Fibonacci or the Thue-Morse morphism, we show that the subshift (calS, T) is sofic, that is, the language of calS is regular.
Ergodic Theory and Dynamical Systems 07/2013; 35(06). DOI:10.1017/etds.2014.5 · 0.78 Impact Factor

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##### Article: Extremal words in morphic subshifts
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ABSTRACT: Given an infinite word X over an alphabet A a letter b occurring in X, and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.
Discrete Mathematics 01/2013; 322(1). DOI:10.1016/j.disc.2014.01.002 · 0.56 Impact Factor
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##### Article: Primitive Words and Lyndon Words in Automatic and Linearly Recurrent Sequences
Daniel Goc · Kalle Saari · Jeffrey Shallit ·
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ABSTRACT: We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length n in a k-automatic sequence is k-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.
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##### Article: Lyndon words and Fibonacci numbers
Kalle Saari ·
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ABSTRACT: It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound log_{2}(n)} + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is log_{phi}(n) + 1, where phi denotes the golden ratio (1 + sqrt{5})/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping L_x that counts the number of Lyndon factors of length at most n in an infinite word x. We show that a recurrent infinite word x is aperiodic if and only if L_x >= L_f, where f is the Fibonacci infinite word, with equality if and only if f is in the shift orbit closure of f.
Journal of Combinatorial Theory Series A 07/2012; 121(1). DOI:10.1016/j.jcta.2013.09.002 · 0.78 Impact Factor
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##### Article: A new geometric approach to Sturmian words
Kaisa Matomäki · Kalle Saari ·
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ABSTRACT: We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.
Theoretical Computer Science 01/2012; 432. DOI:10.1016/j.tcs.2012.01.040 · 0.66 Impact Factor
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##### Article: Avoiding Abelian Powers in Binary Words with Bounded Abelian Complexity
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ABSTRACT: The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerbühler.
International Journal of Foundations of Computer Science 11/2011; 22(4):905--920. DOI:10.1142/S0129054111008489 · 0.30 Impact Factor
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##### Conference Paper: Local Squares, Periodicity and Finite Automata
Mari Huova · Juhani Karhumäki · Aleksi Saarela · Kalle Saari ·
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ABSTRACT: We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on whether the square is that of words, that of abelian words or, as an in between case, that of so called k-abelian words. The above nine variants of the problem are completely solved, and some open problems are addressed in the k-abelian case. Finally, an amazing unavoidability result for 2-abelian squares is obtained.
Rainbow of Computer Science - Dedicated to Hermann Maurer on the Occasion of His 70th Birthday; 01/2011
• ##### Article: Standard Factors of Sturmian Words
Gwénaël Richomme · Kalle Saari · Luca Q. Zamboni ·
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ABSTRACT: Among the various ways to construct a characteristic Sturmian word, one of the most used consists in defining an infinite sequence of prefixes that are standard. Nevertheless in any characteristic word c, some standard words occur that are not prefixes of c. We characterize all standard words occurring in any characteristic word (and so in any Sturmian word) using firstly morphisms, then standard prefixes and finally palindromes.
RAIRO - Theoretical Informatics and Applications 02/2010; 44(1). DOI:10.1051/ita/2010011 · 0.33 Impact Factor
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##### Article: Everywhere -repetitive sequences and Sturmian words
Kalle Saari ·
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ABSTRACT: Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhereα-repetitive sequences. Such a sequence is defined by the property that there exists an integer N≥2 such that every length-N factor has a repetition of order α as a prefix. If each repetition is of order strictly larger than α, then the sequence is called everywhereα+-repetitive. In both cases, the number of distinct minimal α-repetitions (or α+-repetitions) occurring in the sequence is finite.A natural question regarding global regularity is to determine the least number, denoted by M(α), of distinct minimalα-repetitions such that an α-repetitive sequence is not necessarily ultimately periodic. We call the everywhere α-repetitive sequences witnessing this property optimal. In this paper, we study optimal 2-repetitive sequences and optimal 2+-repetitive sequences, and show that Sturmian words belong to both classes. We also give a characterization of 2-repetitive sequences and solve the values of M(α) for 1≤α≤15/7.
European Journal of Combinatorics 01/2010; 31(1):177-192. DOI:10.1016/j.ejc.2009.01.004 · 0.65 Impact Factor
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##### Article: Abelian Complexity in Minimal Subshifts
Gwénaël Richomme · Kalle Saari · Luca Q. Zamboni ·
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ABSTRACT: In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of G. Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerden's Theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words we prove something stronger: For every Sturmian word w and positive integer k, each sufficiently long factor of w begins in an Abelian k-power. Comment: 19 pages
Journal of the London Mathematical Society 11/2009; 83(1). DOI:10.1112/jlms/jdq063 · 0.82 Impact Factor
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##### Article: Abelian Properties of Words (Extended abstract)
Gwénaël Richomme · Kalle Saari · Luca Q. Zamboni ·
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ABSTRACT: We say that two finite words $u$ and $v$ are abelian equivalent if and only if they have the same number of occurrences of each letter, or equivalently if they define the same Parikh vector. In this paper we investigate various abelian properties of words including abelian complexity, and abelian powers. We study the abelian complexity of the Thue-Morse word and the Tribonacci word, and answer an old question of G. Rauzy by exhibiting a class of words whose abelian complexity is everywhere equal to 3. We also investigate abelian repetitions in words and show that any infinite word with bounded abelian complexity contains abelian $k$-powers for every positive integer $k$.
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##### Article: Balance and Abelian Complexity of the Tribonacci word
Gwénaël Richomme · Kalle Saari · Luca Q. Zamboni ·
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ABSTRACT: G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $R^2$, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci word $t$ which is the unique fixed point of $\tau$. We show that $AC(n)\in {3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all factors $U$ and $V$ of $t$ of equal length, and for every letter $a \in {0,1,2}$, the number of occurrences of $a$ in $U$ and the number of occurrences of $a$ in $V$ differ by at most 2. While this result is announced in several papers, to the best of our knowledge no proof of this fact has ever been published. We offer two very different proofs of the 2-balance property of $t$. The first uses the word combinatorial properties of the generating morphism, while the second exploits the spectral properties of the incidence matrix of $\tau$. Comment: 20 pages, 1 figure. This is an extended version of 0904.2872v1
Advances in Applied Mathematics 04/2009; 45(2). DOI:10.1016/j.aam.2010.01.006 · 0.82 Impact Factor
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##### Article: On highly palindromic words
Štěpán Holub · Kalle Saari ·
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ABSTRACT: We study some properties of palindromic (scattered) subwords of binary words. In view of the classical problem on subwords, we show that the set of palindromic subwords of a word characterizes the word up to reversal.Since each word trivially contains a palindromic subword of length at least half of its length–a power of the prevalent letter–we call a word that does not contain any palindromic subword longer than half of its length minimal palindromic. We show that every minimal palindromic word is abelian unbordered, that is, no proper suffix of the word can be obtained by permuting the letters of a proper prefix.We also propose to measure the degree of palindromicity of a word w by the ratio |rws|/|w|, where the word rws is minimal palindromic and rs is as short as possible. We prove that the ratio is always bounded by four, and construct a sequence of words that achieves this bound asymptotically.
Discrete Applied Mathematics 03/2009; 157(5-157):953-959. DOI:10.1016/j.dam.2008.03.039 · 0.80 Impact Factor
• ##### Article: Least Periods of Factors of Infinite Words
James D. Currie · Kalle Saari ·
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ABSTRACT: We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.
RAIRO - Theoretical Informatics and Applications 01/2009; 43(1). DOI:10.1051/ita:2008006 · 0.33 Impact Factor
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##### Article: Standard words and abelian powers in Sturmian words
Gwénaël Richomme · Kalle Saari · Luca Q Zamboni ·
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ABSTRACT: We give three descriptions of the factors of a Sturmian word that are standard words. We also show that all Sturmian words are so-called everywhere abelian k-repetitive for all integers k ≥ 1, that is, all sufficiently long factors have an abelian kth power as a prefix. More precisely, given a Sturmian word t and an integer k, there exist two integers 1 and 2 such that each position in t has an abelian kth power with abelian period 1 or 2 .
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##### Article: Transposition invariant words
Arto Lepistö · Francesco Pappalardi · Kalle Saari ·
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ABSTRACT: We define an operation called transposition on words of fixed length. This operation arises naturally when the letters of a word are considered as entries of a matrix. Words that are invariant with respect to transposition are of special interest. It turns out that transposition invariant words have a simple interpretation by means of elementary group theory. This leads us to investigate some properties of the ring of integers modulo nn and primitive roots. In particular, we show that there are infinitely many prime numbers pp with a primitive root dividing p+1p+1 and infinitely many prime numbers pp without a primitive root dividing p+1p+1. We also consider the orbit of a word under transposition.
Theoretical Computer Science 06/2007; 380(3):377-387. DOI:10.1016/j.tcs.2007.03.029 · 0.66 Impact Factor
• ##### Article: Everywhere alpha-Repetitive Sequences and Sturmian Words
Kalle Saari ·
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ABSTRACT: Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhere α -repetitive sequences, sequences in which every position has an occurrence of a repetition of order α ≥ 1 of bounded length. The number of minimal such repetitions, called minimal α -powers, is then finite. A natural question regarding global regularity is to determine the least number of minimal α-powers such that an α-repetitive sequence is not necessarily ultimately periodic. We solve this question for 1 ≤ α ≤ 17/8. We also show that Sturmian words are among the optimal 2 - and 2 + -repetitive sequences.
• ##### Conference Paper: On the Frequency of Letters in Morphic Sequences.
Kalle Saari ·
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ABSTRACT: A necessary and sufficient criterion for the existence and value of the frequency of a letter in a morphic sequence is given. This is done using a certain incidence matrix associated with the morphic sequence. The characterization gives rise to a simple if-and-only-if condition that all letter frequencies exist.
Computer Science - Theory and Applications, First International Computer Science Symposium in Russia, CSR 2006, St. Petersburg, Russia, June 8-12, 2006, Proceedings; 01/2006

#### Publication Stats

 151 Citations 7.95 Total Impact Points

#### Institutions

• ###### The University of Winnipeg
• Department of Mathematics and Statistics