Families of Eliahou semigroups linked to Farey intervals

We describe new families of Eliahou semigroups, encompassing previous families described by Delgado, Eliahou and Fromentin, and Bras-Amorós. A crucial parameter is a Farey interval associated to the semigroup. We show that these semigroups probably all satisfy Wilf’s conjecture and describe ways to explicitly construct semigroups belonging to these families. This work is based on an exploration of the numerical semigroup tree giving (conjecturally) all Eliahou semigroups of conductor up to 320 thanks to a new way of representing the semigroups and pruning of unwanted branches.