Characterization of some binary words with few squares

Thue proved that the factors occurring infinitely many times in square-free words over {0,1,2} avoiding the factors in {010,212} are the factors of the fixed point of the morphism 0 → 012, 1 → 02, 2 → 1. He similarly characterized square-free words avoiding {010,020} and {121,212} as the factors of two morphic words. In this paper, we exhibit smaller morphisms to define these two square-free morphic words and we give such characterizations for six types of binary words containing few distinct squares

Avoidability of formulas with two variables

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables 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 $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words

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

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

Strict Bounds for Pattern Avoidance

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

Abelian complexity of fixed point of morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1

We study the combinatorics of vtm, a variant of the Thue-Morse word generated by the non-uniform morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1 starting with 0. This infinite ternary sequence appears a lot in the literature and finds applications in several fields such as combinatorics on words; for example, in pattern avoidance it is often used to construct infinite words avoiding given patterns. It has been shown that the factor complexity of vtm, i.e., the number of factors of length n, is Θ(n); in fact, it is bounded by ¹⁰⁄₃n for all n, and it reaches that bound precisely when n can be written as 3 times a power of 2. In this paper, we show that the abelian complexity of vtm, i.e., the number of Parikh vectors of length n, is O(log n) with constant approaching ¾ (assuming base 2 logarithm), and it is Ω(1) with constant 3 (and these are the best possible bounds). We also prove some results regarding factor indices in vtm."F. Blanchet-Sadri and Nathan Fox’s research was supported by the National Science Foundation under Grant No. DMS–1060775." "James D. Currie and Narad Rampersad’s research was supported by NSERC Discovery grants.

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. © 2014 EDP Sciences

Binary words avoiding the pattern AABBCABBA

We show that there are three types of infinite words over the two-letter alphabet {0,1} that avoid the pattern AABBCABBA. These types, P, E0, and E1, differ by the factor complexity and the asymptotic frequency of the letter 0. Type P has polynomial factor complexity and letter frequency $\frac{1}{2}$. Type E0 has exponential factor complexity and the frequency of the letter 0 is at least 0.45622 and at most 0.48684. Type E1 is obtained from type E0 by exchanging 0 and 1