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DOI: 10.4230/LIPIcs.CPM.2019.2

Some Variations on Lyndon Words (Invited Talk)

Authors: Francesco Dolce, Antonio Restivo, and Christophe Reutenauer

Published in: LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

Abstract

In this paper we compare two finite words u and v by the lexicographical order of the infinite words u^omega and v^omega. Informally, we say that we compare u and v by the infinite order. We show several properties of Lyndon words expressed using this infinite order. The innovative aspect of this approach is that it allows to take into account also non trivial conditions on the prefixes of a word, instead that only on the suffixes. In particular, we derive a result of Ufnarovskij [V. Ufnarovskij, Combinatorial and asymptotic methods in algebra, 1995] that characterizes a Lyndon word as a word which is greater, with respect to the infinite order, than all its prefixes. Motivated by this result, we introduce the prefix standard permutation of a Lyndon word and the corresponding (left) Cartesian tree. We prove that the left Cartesian tree is equal to the left Lyndon tree, defined by the left standard factorization of Viennot [G. Viennot, Algèbres de Lie libres et monoïdes libres, 1978]. This result is dual with respect to a theorem of Hohlweg and Reutenauer [C. Hohlweg and C. Reutenauer, Lyndon words, permutations and trees, 2003].

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Francesco Dolce, Antonio Restivo, and Christophe Reutenauer. Some Variations on Lyndon Words (Invited Talk). In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dolce_et_al:LIPIcs.CPM.2019.2,
  author =	{Dolce, Francesco and Restivo, Antonio and Reutenauer, Christophe},
  title =	{{Some Variations on Lyndon Words}},
  booktitle =	{30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-103-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{128},
  editor =	{Pisanti, Nadia and P. Pissis, Solon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.2},
  URN =		{urn:nbn:de:0030-drops-104732},
  doi =		{10.4230/LIPIcs.CPM.2019.2},
  annote =	{Keywords: Lyndon words, Infinite words, Left Lyndon trees, Left Cartesian trees}
}

Document

DOI: 10.4230/LIPIcs.ICALP.2016.124

Anti-Powers in Infinite Words

Authors: Gabriele Fici, Antonio Restivo, Manuel Silva, and Luca Q. Zamboni

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract

In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. From these results, we derive that at every position of an aperiodic uniformly recurrent word start anti-powers of any order. We further show that any infinite word avoiding anti-powers of order 3 is ultimately periodic, and that there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6, and leave open the question whether there exist aperiodic recurrent words avoiding anti-powers of order k for k=4,5.

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Gabriele Fici, Antonio Restivo, Manuel Silva, and Luca Q. Zamboni. Anti-Powers in Infinite Words. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 124:1-124:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fici_et_al:LIPIcs.ICALP.2016.124,
  author =	{Fici, Gabriele and Restivo, Antonio and Silva, Manuel and Zamboni, Luca Q.},
  title =	{{Anti-Powers in Infinite Words}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{124:1--124:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.124},
  URN =		{urn:nbn:de:0030-drops-62599},
  doi =		{10.4230/LIPIcs.ICALP.2016.124},
  annote =	{Keywords: infinite word, anti-power, unavoidable regularity, avoidability}
}

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Documents authored by Restivo, Antonio

Some Variations on Lyndon Words (Invited Talk)

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Anti-Powers in Infinite Words

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