67 research outputs found

    Avoidability index for binary patterns with reversal

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    For every pattern pp over the alphabet {x,y,xR,yR}\{x,y,x^R,y^R\}, we specify the least kk such that pp is kk-avoidable.Comment: 15 pages, 1 figur

    A family of formulas with reversal of high avoidability index

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    We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size and the simple structure of its members. For each k ∈ {4,5}, there are several previously known avoidable formulas (without reversal) of avoidability index k, but they are small in number and they all have rather complex structure.http://dx.doi.org/10.1142/S021819671750024

    On the aperiodic avoidability of binary patterns with variables and reversals

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    In this work we present a characterisation of the avoidability of all unary and binary patterns, that do not only contain variables but also reversals of their instances, with respect to aperiodic infinite words. These types of patterns were studied recently in either more general or particular cases

    Strict Bounds for Pattern Avoidance

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    Cassaigne conjectured in 1994 that any pattern with m distinct variables of length at least 3(2m-1) is avoidable over a binary alphabet, and any pattern with m distinct variables of length at least 2m is avoidable over a ternary alphabet. Building upon the work of Rampersad and the power series techniques of Bell and Goh, we obtain both of these suggested strict bounds. Similar bounds are also obtained for pattern avoidance in partial words, sequences where some characters are unknown

    Growth rate of binary words avoiding xxxRxxx^R

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    Consider the set of those binary words with no non-empty factors of the form xxxRxxx^R. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the existence of upper and lower bounds on the number of such words of length nn, where each of these bounds is asymptotically equivalent to a (different) function of the form Cnlgn+cCn^{\lg n+c}, where CC, cc are constants

    Computing the Partial Word Avoidability Indices of Ternary Patterns

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    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

    Pattern Avoidability with Involution

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    An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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