8,486 research outputs found
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
Doubled patterns are -avoidable
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet if there is no factor of
such that where is a non-erasing morphism. A
pattern is said to be -avoidable if there exists an infinite word over a
-letter alphabet that avoids . A pattern is said to be doubled if no
variable occurs only once. Doubled patterns with at most 3 variables and
patterns with at least 6 variables are -avoidable. We show that doubled
patterns with 4 and 5 variables are also -avoidable
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
Avoiding Patterns in the Abelian Sense
We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ω-words avoiding these patterns.Research of both authors supported by NSERC Operating Grants.https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/avoiding-patterns-in-the-abelian-sense/42148B0781A38A6618A537AAD7D39B8
Singleton mesh patterns in multidimensional permutations
This paper introduces the notion of mesh patterns in multidimensional
permutations and initiates a systematic study of singleton mesh patterns
(SMPs), which are multidimensional mesh patterns of length 1. A pattern is
avoidable if there exist arbitrarily large permutations that do not contain it.
As our main result, we give a complete characterization of avoidable SMPs using
an invariant of a pattern that we call its rank. We show that determining
avoidability for a -dimensional SMP of cardinality is an problem, while determining rank of is an NP-complete problem.
Additionally, using the notion of a minus-antipodal pattern, we characterize
SMPs which occur at most once in any -dimensional permutation. Lastly, we
provide a number of enumerative results regarding the distributions of certain
general projective, plus-antipodal, minus-antipodal and hyperplane SMPs.Comment: Theorem 12 and Conjecture 1 are replaced by a more general Theorem
12; the paper is to appear in JCT
Computing the Partial Word Avoidability Indices of Ternary Patterns
We study pattern avoidance in the context of partial words. The problem of classifying the avoidable binary patterns has been solved, so we move on to ternary and more general patterns. Our results, which are based on morphisms (iterated or not), determine all the ternary patterns' avoidability indices or at least give bounds for them
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