On a conjecture of Wilf

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.Comment: 18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series

On a conjecture of Wilf about the Frobenius number

Given coprime positive integers $a_1 < ...< a_d$, 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 $d \geq \frac{F+1}{F+1-g}$. 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 $d = \frac{F+1}{F+1-g}$

Combinatorial enumeration of weighted Catalan numbers

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 69-70).This thesis is devoted to the divisibility property of weighted Catalan and Motzkin numbers and its applications. In Chapter 1, the definitions and properties of weighted Catalan and Motzkin numbers are introduced. Chapter 2 studies Wilf conjecture on the complementary Bell number, the alternating sum of the Stirling number of the second kind. Congruence properties of the complementary Bell numbers are found by weighted Motkin paths, and Wilf conjecture is partially proved. In Chapter 3, Konvalinka conjecture is proved. It is a conjecture on the largest power of two dividing weighted Catalan number, when the weight function is a polynomial. As a corollary, we provide another proof of Postnikov and Sagan of weighted Catalan numbers, and we also generalize Konvalinka conjecture for a general weight function.by Junkyu An.Ph.D

Semigrups numèrics: semimóduls, sizígies i la seua relació amb la conjetura de Wilf

Treball Final de Grau en Matemàtica Computacional. Codi: MT1030. Curs: 2019/2020Aquest 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