234 research outputs found
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
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 ( [Formula presented] )-power-free binary words, but not over all ( [Formula presented] )+-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 ( [Formula presented] )-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 symmetric square-free ternary words. © 2018 Elsevier B.V
Avoiding Abelian powers in binary words with bounded Abelian complexity
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 -powers for some integer , then its Abelian complexity is
unbounded. This suggests the following question: How frequently do Abelian
-powers occur in a word having bounded Abelian complexity? In particular,
does every uniformly recurrent word having bounded Abelian complexity begin in
an Abelian -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-Hungerb\"uhler.Comment: 16 pages, submitte
Counting occurrences of some subword patterns
We find generating functions the number of strings (words) containing a
specified number of occurrences of certain types of order-isomorphic classes of
substrings called subword patterns. In particular, we find generating functions
for the number of strings containing a specified number of occurrences of a
given 3-letter subword pattern.Comment: 9 page
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
- …