Related papers: Wilf's conjecture for numerical semigroups

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…

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…

Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}) and conductor c = max(N \ S) + 1. Let P be the set of primitive elements of S, and let L be the set of elements of S which are smaller than c. A longstand-ing…

For a numerical semigroup $S \subseteq \mathbb{N}$, let $m,e,c,g$ denote its multiplicity, embedding dimension, conductor and genus, respectively. Wilf's conjecture (1978) states that $e(c-g) \ge c$. As of 2023, Wilf's conjecture has been…

Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that…

For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known…

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 18$. Our algorithm utilizes techniques from polyhedral…

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical…

Let $\CaC\subset \Q^p$ be a rational cone. An affine semigroup $S\subset \CaC$ is a $\CaC$-semigroup whenever $(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\CaC$-semigroups, give a…

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 introduce a new way of counting numerical semigroups, namely by their maximum primitive, and show its relation with the counting of numerical semigroups by their Frobenius number. We show that these two ways of counting are M\"obius…

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…

The aim of this paper is to propose an extension of the Wilf conjecture to semimodules over a numerical semigroup through a new approach toward the solution of the Wilf conjecture on numerical semigroups. The key point is the introduction…

Let S $\subseteq$ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S $\cap$ [0, c -- 1] and W(S) = |P||L| -- c. In 1978, Herbert Wilf asked whether W(S) $\ge$ 0 always holds, a question known…

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,…

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…

To a numerical semigroup $S$, Eliahou associated a number $E(S)$ and proved that numerical semigroups for which the associated number is non negative satisfy Wilf's conjecture. The search for counterexamples for the conjecture of Wilf is…

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…

In this paper we introduce a particular semigroup transform $\mathcal{A}$ that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of not ordinary and not irreducible…

Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1…