Dejean's conjecture holds for n >= 30

We extend Carpi's results by showing that Dejean's conjecture holds for n >= 30.Comment: introductory material added, minor corrections, 6 page

Binary words avoiding xx^Rx and strongly unimodal sequences

In previous work, Currie and Rampersad showed that the growth of the number of binary words avoiding the pattern xxx^R was intermediate between polynomial and exponential. We now show that the same holds for the growth of the number of binary words avoiding the pattern xx^Rx. Curiously, the analysis for xx^Rx is much simpler than that for xxx^R. We derive our results by giving a bijection between the set of binary words avoiding xx^Rx and a class of sequences closely related to the class of "strongly unimodal sequences."Comment: 4 page

Growth rate of binary words avoiding xxxRxxx^R

Consider the set of those binary words with no non-empty factors of the form $xxx^R$. 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 $n$, where each of these bounds is asymptotically equivalent to a (different) function of the form $Cn^{\lg n+c}$, where $C$, $c$ are constants

A family of formulas with reversal of high avoidability index

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