Related papers: Dynamic and Programmatic Analysis of Fibonacci Wor…
Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have…
We determine the order of magnitude of the variance of the Fibonacci partition function. The answer is different to the most naive guess. The proof involves a diophantine system and an inhomogeneous linear recurrence.
In this work, we treat subshifts, defined in terms of an alphabet $A$ and (usually infinite) forbidden list $F$, where the number of $n$-letter words in $F$ has "slow growth rate" in $n$. We show that such subshifts are well-behaved in…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…
We show that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density.
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz…
We present methods for calculating a measure of phonotactic complexity---bits per phoneme---that permits a straightforward cross-linguistic comparison. When given a word, represented as a sequence of phonemic segments such as symbols in the…
We investigate the compression sensitivity [Akagi et al., 2023] of lex-parse [Navarro et al., 2021] for two operations: (1) single character edit and (2) modification of the alphabet ordering, and give tight upper and lower bounds for both…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
This paper begins with a comprehensive overview of combinatorics on words and symbolic dynamics, covering their historical origins, fundamental concepts, and interconnections. Building upon this foundation, we introduce novel mathematical…
The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, approximation theory worked as a…
Finding models for linear-time properties is a central problem in verification and planning. We study the distribution of linear-time models by investigating the density of linear-time properties over the space of ultimately periodic words.…
Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is…
For a prime $p$, let $Z(p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$, the $n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $\zeta(m)$, the density of primes $p$ for…
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 establish several recurrence relations and an explicit formula for V(n), the number of factorizations of the length-n prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In…
We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the…
We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this…
Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the most famous example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to…