Middle divisors and σ\sigma-palindromic Dyck words

Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle divisor of $n$ if $$\sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}.$$ We will prove that there are integers having an arbitrarily large number of $\lambda$-middle divisors. Consider the word $$\langle\! \langle n \rangle\! \rangle_{\lambda} := w_1 w_2 ... w_k \in \{a,b\}^{\ast},$$ given by $$w_i := \left\{ \begin{array}{c l} a & \textrm{if } u_i \in D_n \backslash \left(\lambda D_n\right), \\ b & \textrm{if } u_i \in \left(\lambda D_n\right)\backslash D_n, \end{array} \right.$$ where $D_n$ is the set of divisors of $n$, $\lambda D_n := \{\lambda d: \quad d \in D_n\}$ and $u_1, u_2, ..., u_k$ are the elements of the symmetric difference $D_n \triangle \lambda D_n$ written in increasing order. We will prove that the language $$\mathcal{L}_{\lambda} := \left\{\langle\! \langle n \rangle\! \rangle_{\lambda} : \quad n \in \mathbb{Z}_{\geq 1} \right\}$$ contains Dyck words having an arbitrarily large number of centered tunnels. We will show a connection between both results

A relationship between the ideals of Fq[x,y,x−1,y−1]\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right] and the Fibonacci numbers

Let $C_n(q)$ be the number of ideals of codimension $n$ of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial in $q$ for any fixed value of $n \geq 1$. For $q = \frac{3+\sqrt{5}}{2}$, this combinatorial interpretation of $C_n(q)$ is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears. Let $f_n$ be the $n$-th Fibonacci number (following the convention $f_0 = 0$, $f_1 = 1$). Define the series $$F(t) = \sum_{n \geq 1} f_{2n}\,t^n.$$ We will prove that for each $n \geq 1$, $$C_n\left( \frac{3+\sqrt{5}}{2}\right) = \lambda_n \, \left(f_{2n} \frac{3+\sqrt{5}}{2} - f_{2n-2} \right) ,$$ where the integers $\lambda_n \geq 0$ are given by the following generating function $$ \prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1} \lambda_n\,t^n. $

On a function introduced by Erd\"{o}s and Nicolas

Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$

On prime numbers of the form 2n±k2^n \pm k

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set rational numbers related to a sequence of polynomials $C_n(q)$ introduced by Kassel and Reutenauer [KasselReutenauer]

Factorization of Dyck words and the distribution of the divisors of an integer

In [CaballeroHooleyDelta], we associated a Dyck word $\langle\! \langle n \rangle\! \rangle_{\lambda}$ to any pair $(n, \lambda)$ consisting of an integer $n \geq 1$ and a real number $\lambda > 1$. The goal of the present paper is to show a relationship between the factorization of $\langle\! \langle n \rangle\! \rangle_{\lambda}$ as the concatenation of irreducible Dyck words and the distribution of the divisors of $n$. In particular, we will provide a characterization of $\lambda$-densely divisible numbers (these numbers were introduced in [castryck2014new])

Relative positions of points on the real line and balanced parentheses

Consider a finite set of positive real numbers $S$. For any real number $\lambda > 1$, a Dyck word denoted $\langle\! \langle S \rangle\! \rangle_{\lambda} \in \{a,b\}^{\ast}$, was defined in [CaballeroWords2017] in order to compute Hooley's $\Delta$-function and its generalization. The aim of this paper is to prove that, given a real number $\lambda > 1$, any Dyck word can be expressed as $\langle\! \langle S \rangle\! \rangle_{\lambda}$ for some finite set $S$ of positive real numbers

Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. We study the connection between self-similar algebras with run-length encoding and rational languages. In particular, we provide a curious relationship between the eigenvalues of a sequence of matrices related to a specific self-similar algebra and the smooth words over a 2-letter alphabet. We also consider the language $L(s)$ of words $u$ in $(\Sigma\times \Sigma)^*$ where $\Sigma=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$

Comparing Tag Scheme Variations Using an Abstract Machine Generator

In this paper we study, in the context of a WAM-based abstract machine for Prolog, how variations in the encoding of type information in tagged words and in their associated basic operations impact performance and memory usage. We use a high-level language to specify encodings and the associated operations. An automatic generator constructs both the abstract machine using this encoding and the associated Prolog-to-byte code compiler. Annotations in this language make it possible to impose constraints on the final representation of tagged words, such as the effectively addressable space (fixing, for example, the word size of the target processor /architecture), the layout of the tag and value bits inside the tagged word, and how the basic operations are implemented. We evaluate large number of combinations of the different parameters in two scenarios: a) trying to obtain an optimal general-purpose abstract machine and b) automatically generating a specially-tuned abstract machine for a particular program. We conclude that we are able to automatically generate code featuring all the optimizations present in a hand-written, highly-optimized abstract machine and we canal so obtain emulators with larger addressable space and better performance

Las tres muertes del Mariscal Sucre

pags. 129-15