Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.25051

Families of Eliahou semigroups linked to Farey intervals

Axel Bacher
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe a way to explore the numerical semigroup tree to find all Eliahou semigroups of conductor up to 200 (and conjecturally up to 400) thanks to a new way of representing the semigroups and pruning off unwanted branches. This proves that Wilf's conjecture holds for conductor up to 200. Based on this list, we describe new classes of numerical semigroups, with a Farey interval as a crucial parameter. We show that these semigroups probably all satisfy Wilf's conjecture. We explicitly construct many families of Eliahou semigroups belonging to these classes, encompassing families described by Delgado, Eliahou and Fromentin, and Bras-Amorós.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Axel Bacher. 2026-08-27. Families of Eliahou semigroups linked to Farey intervals. https://arxiv.org/abs/2604.25051

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗