Related papers: Dynamic and Programmatic Analysis of Fibonacci Wor…

We present the new empirical parameter $f_c$, the most probable usage frequency of a word in a language, computed via the distribution of documents over frequency $x$ of the word. This parameter allows for filtering the core lexicon of a…

This paper is a sharp and focussed exploration of the Fibonacci substitution and the mathematical entity it gives rise to, the Fibonacci word. Our investigations are both of an algebraic and a geometric nature. Indeed, it is the combination…

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this…

We define the notion of circular words, then consider on such words a constraint derived from the Fibonacci condition. We give several results on the structure of these circular words, then mention possible applications to various…

As is the case of many signals produced by complex systems, language presents a statistical structure that is balanced between order and disorder. Here we review and extend recent results from quantitative characterisations of the degree of…

Both Fibonacci and Lucas numbers can be described combinatorially in terms of 0-1 strings without consecutive ones. In the present article we explore the occupation numbers as well as the correlations between various positions in the…

We determine the maximal length of the period of a periodic word defined by $n$ restrictions. It happens to be the corresponding Fibonacci number.

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…

Let $D^+$ be the first octant of the Euclidean space and consider the integral cube grid $G$ in $D^+$. The intersections of each line with $G$ form an infinite sequence of three letters which can be considered as an extension of well-known…

For each positive integer $k$, let $\mathscr{A}_k$ be the set of all positive integers $n$ such that $\gcd(n, F_n) = k$, where $F_n$ denotes the $n$th Fibonacci number. We prove that the asymptotic density of $\mathscr{A}_k$ exists and is…

Current lexical simplification approaches rely heavily on heuristics and corpus level features that do not always align with human judgment. We create a human-rated word-complexity lexicon of 15,000 English words and propose a novel neural…

Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials.…

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…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which…

We investigate combinatorial properties of a kind of insets we defined in an earlier paper, interpreting them now in terms of restricted ternary words. This allows us to give new combinatorial interpretations of a number of known integer…

We consider a random generalisation of the classical Fibonacci substitution. The substitution we consider is defined as the rule mapping $\mathtt{a}\mapsto \mathtt{baa}$ and $\mathtt{b} \mapsto \mathtt{ab}$ with probability $p$ and…

It is proved that the asymptotic average eccentricity and the asymptotic average degree of Fibonacci cubes and Lucas cubes are $(5+\sqrt 5)/10$ and $(5-\sqrt 5)/5$, respectively. A new labeling of the leaves of Fibonacci trees is introduced…

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words…

In the area of pattern avoidability the central role is played by special words called Zimin patterns. The symbols of these patterns are treated as variables and the rank of the pattern is its number of variables. Zimin type of a word $x$…