146 research outputs found
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -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 is -rich. We
also calculate the factor complexity of .Comment: 11 page
Subword complexity and power avoidance
We begin a systematic study of the relations between subword complexity of
infinite words and their power avoidance. Among other things, we show that
-- the Thue-Morse word has the minimum possible subword complexity over all
overlap-free binary words and all -power-free binary words, but not
over all -power-free binary words;
-- the twisted Thue-Morse word has the maximum possible subword complexity
over all overlap-free binary words, but no word has the maximum subword
complexity over all -power-free binary words;
-- if some word attains the minimum possible subword complexity over all
square-free ternary words, then one such word is the ternary Thue word;
-- the recently constructed 1-2-bonacci word has the minimum possible subword
complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
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