On a question of Eliahou and a conjecture of Wilf

TL;DR

This paper constructs infinite families of numerical semigroups with any given Eliahou number, all satisfying Wilf's conjecture, thus expanding the known classes of semigroups for which the conjecture holds.

Contribution

It explicitly constructs infinite families of numerical semigroups with any Eliahou number, all satisfying Wilf's conjecture, providing new evidence supporting the conjecture.

Findings

Infinite families of semigroups with any Eliahou number are constructed.

All constructed semigroups satisfy Wilf's conjecture.

The work extends the known classes of semigroups supporting Wilf's conjecture.

Abstract

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 therefore reduced to semigroups which have an associated negative Eliahou number. Eliahou mentioned $5$ numerical semigroups whose Eliahou number is $-1$. The examples were discovered by Fromentin who observed that these are the only ones with negative Eliahou number among the over $10^{13}$ numerical semigroups of genus up to $60$. We prove here that for any integer $n$ there are infinitely many numerical semigroups S such that $E(S)=n$, by explicitly giving families of such semigroups. We prove that all the semigroups in these families satisfy Wilf's conjecture, thus providing not previously known examples of semigroups for which the conjecture holds.