2 : 2 Some Variations on Lyndon

In this paper we compare two finite words u and v by the lexicographical order of the infinite words u and v. 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]. 2012 ACM Subject Classification Mathematics of computing → Combinatorics on words

[1]  Antonio Restivo,et al.  Block Sorting-Based Transformations on Words: Beyond the Magic BWT , 2018, DLT.

[2]  Kazuya Tsuruta,et al.  The "Runs" Theorem , 2014, SIAM J. Comput..

[3]  Kazuya Tsuruta,et al.  A new characterization of maximal repetitions by Lyndon trees , 2015, SODA.

[4]  Antonio Restivo,et al.  Sorting conjugates and Suffixes of Words in a Multiset , 2014, Int. J. Found. Comput. Sci..

[5]  Wojciech Rytter,et al.  The maximal number of cubic runs in a word , 2012, J. Comput. Syst. Sci..

[6]  Antonio Restivo,et al.  A bijection between words and multisets of necklaces , 2012, Eur. J. Comb..

[7]  Manfred Kufleitner.  On Bijective Variants of the Burrows-Wheeler Transform , 2009, Stringology.

[8]  Antonio Restivo,et al.  An extension of the Burrows-Wheeler Transform , 2007, Theor. Comput. Sci..

[9]  Maxime Crochemore,et al.  Fast parallel Lyndon factorization with applications , 1995, Mathematical systems theory.

[10]  Christophe Reutenauer,et al.  Lyndon words, permutations and trees , 2003, Theor. Comput. Sci..

[11]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[12]  Christian Choffrut,et al.  Combinatorics of Words , 1997, Handbook of Formal Languages.

[13]  V. A. Ufnarovskij.  Combinatorial and Asymptotic Methods in Algebra , 1995 .

[14]  Ira M. Gessel,et al.  Counting Permutations with Given Cycle Structure and Descent Set , 1993, J. Comb. Theory A.

[15]  Jean Pierre Duval,et al.  Factorizing Words over an Ordered Alphabet , 1983, J. Algorithms.

[16]  G. Viennot.  Algèbres de Lie Libres et Monoïdes Libres , 1978 .

[17]  G. Bergman.  Centralizers in free associative algebras , 1969 .

[18]  R. Lyndon.  On Burnside’s problem , 1954 .