25 research outputs found
Growth rate of binary words avoiding
Consider the set of those binary words with no non-empty factors of the form
. 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
, where each of these bounds is asymptotically equivalent to a (different)
function of the form , where , are constants
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
Avoidability index for binary patterns with reversal
For every pattern over the alphabet , we specify the
least such that is -avoidable.Comment: 15 pages, 1 figur
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
On the aperiodic avoidability of binary patterns with variables and reversals
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
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
Is Büchi's theorem useful for you? (for an audience of logicians)
Almost a century ago, Presburger showed that the first order theory of the natural numbers with addition is decidable. Following the work of B\"uchi in 1960, this result still holds when adding a function to the structure, where is the largest power of diving . In particular, this leads to a logical characterization of the -automatic sequences.
During the last few years, many applications of this result have been considered in combinatorics on words, mostly by J. Shallit and his coauthors.
In this talk, we will present this theorem of B\"uchi where decidability relies on finite automata. Then we will review some results about automatic sequences or morphic words that can be proved automatically (i.e., the proof is carried on by an algorithm). Finally, we will sketch the limitation of this technique. With a single line formula, one can prove automatically that the Thue-Morse word has no overlap but, hopefully, not all the combinatorial properties of morphic words can be derived in this way.
We will not assume any background in combinatorics on words from the audience
Annual Report of the Board of Regents of the Smithsonian Institution, showing the operations, expenditures, and condition of the Institution for the year ending June 30, 1887
Annual Report of the Smithsonian Institution. [2581-2582] Research related to the American Indian