Some rare numerical semigroups

A numerical semigroup $S$ is a co-finite submonid of the monoid of the non-negative integers, under addition. The number of positive integers that do not belong to $S$ is called the \textit{genus} of $S$. Two elements of great importance in $S$ are: its smallest positive integer, called the \emph{multiplicity} of $S$, and the smallest integer after which all the integers belong to $S$, called the \textit{conductor} of $S$ and denoted $c$.

Semigroups whose conductor is more than three times bigger than the multiplicity are rare in the following sense: denote the number of numerical semigroups of genus $g$ by $n_g$ and denote the number of those having conductor greater than three times the multiplicity by $t_g$, the quotient $t_g/n_g$ aproaches $1$ as $g$ grows, as shown by Zhai.

Eliahou proved that all the numerical semigroups whose conductor is not bigger than three times the multiplicity satisfy Wilf's conjecture. He assigned to each numerical semigroup a number (that depends on the conductor, number of elements less than the conductor, and minimal generators), which we call the Eliahou number, and proved that all the semigroups under consideration have non-negative Eliahou number. Furthermore, numerical semigroups with non-negative Eliahou number satisfy Wilf's conjecture.

Numerical semigroups whose Eliahou number is negative are even rarer than those with conductor bigger than three times the multiplicity. Among the over $10^{13}$ numerical semigroups of genus up to $60$ there are only $5$ with negative Eliahou number (and this number is $-1$).

In this seminar we will give, for each integer, an infinite family of numerical semigroups whose Eliahou number is the given integer. All the semigroups in the families given satisfy Wilf's conjecture.