scholarly journals Wilf’s conjecture and Macaulay’s theorem

2018 ◽  
Vol 20 (9) ◽  
pp. 2105-2129 ◽  
Author(s):  
Shalom Eliahou
Keyword(s):  
Wilf’S Conjecture

Complementary Bell Numbers: Arithmetical Properties and Wilf’s Conjecture

2013 ◽  
pp. 23-56
Author(s):  
Tewodros Amdeberhan ◽  
Valerio De Angelis ◽  
Victor H. Moll
Keyword(s):  
Bell Numbers ◽  
Wilf’S Conjecture

Compositions of a numerical semigroup

2019 ◽  
Vol 29 (5) ◽  
pp. 345-350
Author(s):  
Ze Gu
Keyword(s):  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Apéry Set ◽  
Wilf’S Conjecture

Abstract Given a numerical semigroup S, a nonnegative integer a and m ∈ S ∖ {0}, we introduce the set C(S, a, m) = {s + aw(s mod m) | s ∈ S}, where {w(0), w(1), ⋯, w(m – 1)} is the Apéry set of m in S. In this paper we characterize the pairs (a, m) such that C(S, a, m) is a numerical semigroup. We study the principal invariants of C(S, a, m) which are given explicitly in terms of invariants of S. We also characterize the compositions C(S, a, m) that are symmetric, pseudo-symmetric and almost symmetric. Finally, a result about compliance to Wilf’s conjecture of C(S, a, m) is given.

scholarly journals Near-misses in Wilf’s conjecture

2018 ◽  
Vol 98 (2) ◽  
pp. 285-298
Author(s):  
Shalom Eliahou ◽  
Jean Fromentin
Keyword(s):  
Near Misses ◽  
Wilf’S Conjecture

scholarly journals An extension of Wilf’s conjecture to affine semigroups

2017 ◽  
Vol 96 (2) ◽  
pp. 396-408 ◽  
Author(s):  
J. I. García-García ◽  
D. Marín-Aragón ◽  
A. Vigneron-Tenorio
Keyword(s):  
Affine Semigroups ◽  
Wilf’S Conjecture

scholarly journals A Graph-Theoretic Approach to Wilf's Conjecture

10.37236/9106 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Shalom Eliahou
Keyword(s):  
Graph Theory ◽  
Numerical Semigroup ◽  
Numerical Semigroups ◽  
Theoretic Approach ◽  
Primitive Elements ◽  
Graph Theoretic ◽  
Graph Theoretic Approach ◽  
Wilf’S Conjecture ◽  
Open Question

Let $S \subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m = \min(S \setminus \{0\})$ and conductor $c=\max(\mathbb{N} \setminus S)+1$. 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 longstanding open question by Wilf in 1978 asks whether the inequality $|P||L| \ge c$ always holds. Among many partial results, Wilf's conjecture has been shown to hold in case $|P| \ge m/2$ by Sammartano in 2012. Using graph theory in an essential way, we extend the verification of Wilf's conjecture to the case $|P| \ge m/3$. This case covers more than $99.999\%$ of numerical semigroups of genus $g \le 45$.

scholarly journals Wilf’s conjecture for numerical semigroups with large second generator

2020 ◽  
pp. 2150197
Author(s):  
Dario Spirito
Keyword(s):  
Quadratic Function ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Wilf’S Conjecture

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.

scholarly journals Wilf’s conjecture in fixed multiplicity

2020 ◽  
Vol 30 (04) ◽  
pp. 861-882
Author(s):  
Winfried Bruns ◽  
Pedro García-Sánchez ◽  
Christopher O’Neill ◽  
Dane Wilburne
Keyword(s):  
Automorphism Group ◽  
Polyhedral Cone ◽  
New Method ◽  
Numerical Semigroups ◽  
Polyhedral Geometry ◽  
Finite Poset ◽  
Wilf’S Conjecture ◽  
Multiplicity Formula

We give an algorithm to determine whether Wilf’s conjecture holds for all numerical semigroups with a given multiplicity [Formula: see text], and use it to prove Wilf’s conjecture holds whenever [Formula: see text]. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf’s conjecture via a combinatorially flavored game played on the elements of a certain finite poset.

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