Related papers: A characterization of balanced episturmian sequenc…
We define a family of natural decompositions of Sturmian words in Christoffel words, called *reversible Christoffel* (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always)…
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under…
We introduce the notion of $\alpha$-numbers and formal intercept of sturmian words, and derive from this study general factorisations formula for sturmian words. Sturmian words are defined as infinite words with lowest unbound complexity,…
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct…
We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering…
Stanley, building on work of Stern, defined an array of numbers by the recurrence $s(n, 2k) = s(n-1, k)$, $s(n, 2k+1) = s(n-1, k) + s(n-1, k+1)$. Stanley showed that, for each positive integer $r$, the sequence $s_n^r:= \sum_k s(n,k)^r$…
In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with factors from a…
An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms…
We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically…
Let $k\ge 2$. We prove that the characteristic sequence of a regular language over a $k$-letter alphabet is $k$-automatic. More generally, if $t\ge 2$ and $t,k$ are multiplicatively dependent, we show that the characteristic sequence of a…
Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired…
We prove that episturmian words and Arnoux-Rauzy sequences can be characterized using a local balance property. We also give a new characterization of epistandard words and show that the set of finite words that are not factors of an…
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A…
A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and…
We say a string has a cadence if a certain character is repeated at regular intervals, possibly with intervening occurrences of that character. We call the cadence anchored if the first interval must be the same length as the others. We…
We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes…
Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive…
This paper presents a novel approach to address the constrained coding challenge of generating almost-balanced sequences. While strictly balanced sequences have been well studied in the past, the problem of designing efficient algorithms…
Each acyclic graph, and more generally, each acyclic orientation of the graph associated to a Cartan matrix, allows to define a so-called frise; this is a collection of sequences over the positive natural numbers, one for each vertex of the…
We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and…