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Wilf’s conjecture for numerical semigroups

Prove or disprove that every numerical semigroup S satisfies the inequality (F(S) + 1 − g(S)) / (F(S) + 1) ≥ 1 / e(S), where e(S) denotes the embedding dimension, F(S) the Frobenius number, and g(S) the genus.

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Background

The paper introduces three key invariants of a numerical semigroup S: the embedding dimension e(S), the Frobenius number F(S), and the genus g(S). A long-standing central problem in the field is Wilf’s conjecture, which asserts a universal inequality relating these three invariants for all numerical semigroups.

This conjecture is a major open problem in the theory of numerical semigroups and motivates the paper of typical behaviors of these invariants under random models, which is the main topic of the paper.

References

A central problem in numerical semigroups is to prove or disprove Wilf's conjecture which is that every numerical semigroup $$ satisfies the inequality \frac{\F[] + 1 - \g()} {\F[]+1} \geq \frac{1}{\e()} involving all three parameters. That is, the reciprocal of the embedding dimension is conjectured to be a lower bound on the density of the segment of $$ between 0 and the Frobenius number.

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups (2411.13767 - Bogart et al., 21 Nov 2024) in Section 1: Introduction