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We study the extended Frobenius problem for sequences of the form {fa+fn}n∈N, where {fn}n∈N is the Fibonacci sequence and fa is a Fibonacci number. As a consequence of this study, we show that the family of numerical semigroups associated with these sequences satisfies Wilf’s conjecture., Universidad de Granada / CBUA, Junta de Andalucía (ProyExcel_00868), Junta de Andalucía Grant Number FQM-343.

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, Garc\'ia-S\'anchez, O'Neill and Wilburne, we apply this algorithm to prove Wilf's conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf's conjecture., Comment: 17 pages including long examples and computational data; v2: added section 6 on Wilf's conjecture for numerical semigroups of multiplicity 19, added additional computational data

Neste trabalho sera apresentada a chamada Conjectura de Wilf, que questiona sobre uma relacao entre invariantes de semigrupos numericos. Partindo de exemplos de familias de semigrupos numericos conhecidas que satisfazem a conjectura, sera apresentada a abordagem de Eliahou [5] que corrobora com a validade da conjectura atraves de ferramentas desenvolvidas a partir da profundidade. Tal estudo contribui significativamente com metodos de abordagem da conjectura e determina um novo invariante a se estudar a respeito de semigrupos numericos: o numero de Eliahou. Serao apresentados alguns conceitos e propriedades basicas sobre semigrupos numericos, assim como os pre-requisitos para o metodo de Eliahou para provar que semigrupos numericos com profundidade q ≤ 3 satisfazem a conjectura de Wilf. In this work, it will be presented the so-called Wilf’s conjecture, which asks about a relation between invariants of numerical semigroups. Starting with examples of known families of numerical semigroups satisfying the conjecture, it will be shown the Eliahou’s approach [5] that corroborates to the validity of the conjecture through tools from the depth of a numerical semigroup. This result greatly contributes with the methods for the conjecture and determines a new invariant to be studied in numerical semigroups: the Eliahou number. It will be presented some basic concepts and properties about numerical semigroups, as well as as the prerequisites for the method of Eliahou to prove that numerical semigroups with depth q ≤ 3 satisfy the Wilf’s conjecture. Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq

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Both authors are supported by the Project MTM2017-84890-P (funded by Ministerio de Economía, Industria y Competitividad and Fondo Europeo de Desarrollo Regional FEDER) and by the Junta de Andalucía Grant Number FQM-343. The authors would like to thank the referee for useful comments and suggestions that have helped us to improve the manuscript., 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 number is F and which have exactly k second kind gaps., European Regional Development Fund, Ministerio de Economía, Industria y Competitividad, Gobierno de España, Junta de Andalucía FQM-343

The first author was partially supported by MTM-2017-84890-P and by Junta de Andalucia group FQM-343. The second author is supported by the project FCT PTDC/MAT/73544/2006). 2010 Mathematics Subject Classification: 20M14, 11D07., We define the concentration of a numerical semigroup S as C(S) = max {next(S)(s) - s vertical bar s is an element of S\{0}} wherein next(S)(s) = min {x is an element of S vertical bar s < x}. In this paper, we study the class of numerical semigroups with concentration 2. We give algorithms to calculate the whole set of this class of semigroups with given multiplicity, genus or Frobenius number. Separately, we prove that this class of semigroups verifies the Wilf's conjecture., project FCT (Fundacao para a Ciencia e a Tecnologia) PTDC/MAT/73544/2006, Junta de Andalucia FQM-343 MTM-2017-84890-P

Treball Final de Grau en Matemàtica Computacional. Codi: MT1030. Curs: 2019/2020 Aquest document recull l’estada en pr`actiques i el projecte final de grau de l’assignatura MT1030 – Pràctiques Externes i Projecte de Final de Grau del grau en Matem`atica Computacional en dues parts diferenciades. En la primera part del treball es descriu l’estada en pr`actiques en l’empresa ArkerLabs on s’ha aplicat la teoria de cues per desenvolupar un algoritme d’estimaci´o del temps d’espera en una cua per implementar-lo en el programari Inqueue. En la segona part del treball, estudiem les generalitats i resultats m´es bàsics dels semigrups numèrics. A, m´es, descrivim els conceptes de semimódul, sizígia i la seua relació per a semigrups numèrics generats minimalment per dos elements. Finalment, generalitzem aquests conceptes i, a través de proves numèriques calculades per programes dissenyats per a aquest treball, donem alguns resultats que relacionen la conjectura de Wilf amb l’estudi dels semim´oduls de certs semigrups numèrics. This document presents the external work placement and bachelor’s degree final project from the subject MT1030 – External Work Placement and Bachelor’s Degree Final Project from the bachelo’s degree in Computational Mathematics in two differentiated parts. In the first part of this paper, it is explained the external work placement at the company ArkerLabs. During this external work placement, it was used the queue theory in order to develop an algorithm to estimate the waiting time in a queue to implement itself in the software Inqueue. En the second part of this paper, we study the basic concepts and most important results regarding the numerical semigroups. Furthermore, we describe the concepts of semimodule, syzygy and their relationship in numerical semigroups minimally generated by two elements. Finally, we generalize these concepts and, through numerical tests given by programmes designed for this paper, we present some results which relate the Wilf’s conjecture with the study of semimodules of some numerical semigroups.

Producción Científica, Let C ⊂ Qp be a rational cone. An affine semigroup S ⊂ C is a C-semigroup whenever (C \ S) ∩ Np has only a finite number of elements. In this work, we study the tree of C-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to C- semigroups and give some families of C-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by C-semigroups.

The thesis is made up of two parts. We study in the first part Wilf’s conjecture for numerical semigroups. We give an equivalent form of Wilf’s conjecture in terms of the Apéry set, embedding dimension and multiplicity of a numerical semigroup. We also give an affirmative answer for the conjecture in certain cases. In the second part, we consider a class of almost arithmetic numerical semigroups and give for this class of semigroups explicit formulas for the Apéry set, the Frobenius number, the genus and the pseudo-Frobenius numbers. We also characterize the symmetric (resp. pseudo-symmetric) numerical semigroups for this class of numerical semigroups.; Cette thèse est composée de deux parties. Nous étudions dans la première la conjecture de Wilf pour les semi-groupes numériques et la résolvons dans certains cas. Dans la seconde nous considérons une classe de semi-groupes presque arithmétiques et donnons pour ces semi-groupes des formules explicites pour la base d’Apéry, le nombre de Frobenius, et les nombres de pseudo-Frobenius. Nouscaractérisons aussi ceux qui sont symétriques (resp. pseudo-symétriques).

We study Wilf’s conjecture for numerical semigroups [Formula: see text] such that the second least generator [Formula: see text] of [Formula: see text] satisfies [Formula: see text], where [Formula: see text] is the conductor and [Formula: see text] the multiplicity of [Formula: see text]. In particular, we show that for these semigroups Wilf’s conjecture holds when the multiplicity is bounded by a quadratic function of the embedding dimension.

The thesis is made up of two parts. We study in the first part Wilf’s conjecture for numerical semigroups. We give an equivalent form of Wilf’s conjecture in terms of the Apéry set, embedding dimension and multiplicity of a numerical semigroup. We also give an affirmative answer for the conjecture in certain cases. In the second part, we consider a class of almost arithmetic numerical semigroups and give for this class of semigroups explicit formulas for the Apéry set, the Frobenius number, the genus and the pseudo-Frobenius numbers. We also characterize the symmetric (resp. pseudo-symmetric) numerical semigroups for this class of numerical semigroups., Cette thèse est composée de deux parties. Nous étudions dans la première la conjecture de Wilf pour les semi-groupes numériques et la résolvons dans certains cas. Dans la seconde nous considérons une classe de semi-groupes presque arithmétiques et donnons pour ces semi-groupes des formules explicites pour la base d’Apéry, le nombre de Frobenius, et les nombres de pseudo-Frobenius. Nouscaractérisons aussi ceux qui sont symétriques (resp. pseudo-symétriques).

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series., 18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series A

Given coprime positive integers $$a_1 < \cdots < a_d$$ , the Frobenius number $$F$$ is the largest integer which is not representable as a non-negative integer combination of the $$a_i$$ . Let $$g$$ denote the number of all non-representable positive integers: Wilf conjectured that $$d \ge \frac{F +1}{F+1-g}$$ . We prove that for every fixed value of $$ \lceil \frac{a_1}{d} \rceil $$ the conjecture holds for all values of $$a_1$$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $$d = \frac{F+1}{F+1-g}$$ .