Related papers: Abelian complexity and Abelian co-decomposition
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…
We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite…
Two finite words $u$ and $v$ are called Abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of (the shift orbit closure of) an infinite word $\mathbf{x}$ is the set…
We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete…
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…
In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the…
In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the…
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $\Theta(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square…
Fici and Saarela ([2]) conjectured that a binary word of length n contains at least $\lfloor n/4 \rfloor$ abelian squares. We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the…
An abelian anti-power of order $k$ (or simply an abelian $k$-anti-power) is a concatenation of $k$ consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion…
We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence.
We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word…
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on…
Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting…
It is known that there are infinite words over finite alphabets with Abelian repetition threshold arbitrarily close to 1; however, the construction previously used involves huge alphabets. In this note we give a short cyclic morphism…
In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the…
We fully classify automatic sequences $a$ over a finite alphabet $\Omega$ with the property that each word over $\Omega$ appears is $a$ along an arithmetic progression. Using the terminology introduced by Avgustinovich, Fon-Der-Flaass and…
In this paper, we introduce a variation of the factor complexity, called the $N$-factor complexity, which allows us to characterize the complexity of sequences on an infinite alphabet. We evaluate precisely the $N$-factor complexity for the…
We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of…