122 research outputs found
Every Binary Pattern of Length Greater Than 14 Is Abelian-2-Avoidable
We show that every binary pattern of length greater than 14 is abelian-2-avoidable. The best known upper bound on the length of abelian-2-unavoidable binary pattern was 118, and the best known lower bound is 7.
We designed an algorithm to decide, under some reasonable assumptions, if a morphic word avoids a pattern in the abelian sense. This algorithm is then used to show that some binary patterns are abelian-2-avoidable. We finally use this list of abelian-2-avoidable pattern to show our result. We also discuss the avoidability of binary patterns on 3 and 4 letters
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
A Note on Efficient Computation of All Abelian Periods in a String
We derive a simple efficient algorithm for Abelian periods knowing all
Abelian squares in a string. An efficient algorithm for the latter problem was
given by Cummings and Smyth in 1997. By the way we show an alternative
algorithm for Abelian squares. We also obtain a linear time algorithm finding
all `long' Abelian periods. The aim of the paper is a (new) reduction of the
problem of all Abelian periods to that of (already solved) all Abelian squares
which provides new insight into both connected problems
Pattern avoidance: themes and variations
AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems
Avoidability of long -abelian repetitions
We study the avoidability of long -abelian-squares and -abelian-cubes
on binary and ternary alphabets. For , these are M\"akel\"a's questions.
We show that one cannot avoid abelian-cubes of abelian period at least in
infinite binary words, and therefore answering negatively one question from
M\"akel\"a. Then we show that one can avoid -abelian-squares of period at
least in infinite binary words and -abelian-squares of period at least 2
in infinite ternary words. Finally we study the minimum number of distinct
-abelian-squares that must appear in an infinite binary word
Conferences WORDS, years 1997-2017: Open Problems and Conjectures
International audienceIn connection with the development of the field of Combinatorics on Words, we present a list of open problems and conjectures which were stated in the context of the eleven international meetings WORDS, which held from 1997 to 2017
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
The avoidability of binary patterns by binary cube-free words is investigated
and the exact bound between unavoidable and avoidable patterns is found. All
avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the
growth rates of the avoiding languages are studied. All such languages, except
for the overlap-free language, are proved to have exponential growth. The exact
growth rates of languages avoiding minimal avoidable patterns are approximated
through computer-assisted upper bounds. Finally, a new example of a
pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons
Days 2012
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