Related papers: A ternary square-free sequence avoiding factors eq…
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…
The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. A word (pattern) over a finite set is said to be unavoidable if, for all but finitely many words, there exists a morphism mapping…
In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern…
In [BabStein] Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous…
We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.
Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an…
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…
We find finite-state recurrences to enumerate the words on the alphabet $[n]^r$ which avoid the patterns 123 and $1k(k-1)\dots2$, and, separately, the words which avoid the patterns 1234 and $1k(k-1)\dots2$.
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…
The set of all avoidable patterns in n or fewer letters can be avoided on an alphabet with 2(n+2) letters.
We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We…
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular…
In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules.…
A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantov\'a and Starosta (Discrete Math. 313 (2013)) proved…
In 2008, Kurosaki gave a new construction of a (bi-)infinite squarefree word over three letters. We show that in fact Kurosaki's word avoids 7/4+-powers, which, as shown by Dejean, is optimal over a 3-letter alphabet.
The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an…
There are several versions of permutation pattern avoidance that have arisen in the literature, and some known examples of two different types of pattern avoidance coinciding. In this paper, we examine barred patterns and vincular patterns.…
We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider…
The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence.…
In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate…