A graph-theoretic approach to Wilf's conjecture
Combinatorics
2021-08-19 v1
Abstract
Let S 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 open question by Wilf in 1978 asks whether the inequality |P||L| c always holds. Among many partial results, Wilf's conjecture has been shown to hold in case |P| 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| m/3. This case covers more than 99.999% of numerical semigroups of genus g 45.
Cite
@article{arxiv.1909.03699,
title = {A graph-theoretic approach to Wilf's conjecture},
author = {Shalom Eliahou},
journal= {arXiv preprint arXiv:1909.03699},
year = {2021}
}