Related papers: On counting numerical semigroups by maximum primit…
We present an algorithm to explore various properties of the numerical semigroups with a given maximum primitive. In particular, we count the number of such numerical semigroups and verify that there is no counterexample to Wilf's…
We define the concentration of a numerical semigroup $S$ as $\mathsf{C}(S)=\max \left\{\text{next}_S(s)-s ~|~ s\in S \backslash \{0\}\right\}$ wherein $\text{next}_S(s)=\min\left\{x \in S ~|~ s<x\right\}$. In this paper, we study the class…
We give an affirmative answer to Wilf's conjecture for numerical semigroups satisfying 2 \nu \geq m, where \nu and m are respectively the embedding dimension and the multiplicity of a semigroup. The conjecture is also proved when m \leq 8…
In 1990, Backelin showed that the number of numerical semigroups with Frobenius number $f$ approaches $C_i \cdot 2^{f/2}$ for constants $C_0$ and $C_1$ depending on the parity of $f$. In this paper, we generalize this result to semigroups…
This paper aims to contribute to validate, for numerical semigroups of reasonably large genus, the so-called Conjecture of Wilf. There is no counter-example for the conjecture among the over 3*10^{10} numerical semigroups of genus up to 60,…
Let $S\subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m$, embedding dimension $\nu$ and conductor $c=f+1=qm-\rho$ for some $q,\rho\in\mathbb{N}$ with $\rho<m$. Let Ap$(S,m) = \{w\_0<w_1 < \ldots < w_{m-1}\}$ be the Ap\'ery…
For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of, seed, by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative…
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius…
We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius…
Wilf Conjecture on numerical semigroups is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that the Wilf inequality is preserved under…
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup…
The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of…
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of…
We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the…
Let $0<\lambda\leq1$, $\lambda\notin\left\{\frac24, \frac27, \frac2{10}, \frac2{13}, \ldots\right\}$, be a real and $p$ a prime number, with $[p,p+\lambda p]$ containing at least two primes. Denote by $f_\lambda(p)$ the largest integer…
We study Wilf's conjecture for numerical semigroups $S$ such that the second least generator $a_2$ of $S$ satisfies $a_2>\frac{c(S)+\mu(S)}{3}$, where $c(S)$ is the conductor and $\mu(S)$ the multiplicity of $S$. In particular, we show that…
We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical…
Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the…
We give an estimate of the minimal positive value of the Wilf function of a numerical semigroup in terms of its concentration. We describe necessary conditions for a numerical semigroup to have negative Eliahou number in terms of its…
A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first…