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Wilf's conjecture for numerical semigroups

Let Λ\Lambda be a numerical semigroup, meaning a cofinite subsemigroup of the non-negative integers. Let e(Λ)e(\Lambda) be the size of its minimal generating set, let c(Λ)=max⁡(N∖Λ)+1c(\Lambda)=\max(\mathbb{N}\setminus\Lambda)+1 be its conductor, and let c′(Λ)=∣{λ∈Λ∣λ<c(Λ)}∣c'(\Lambda)=\left|\{\lambda\in\Lambda\mid \lambda<c(\Lambda)\}\right|. Wilf's conjecture.

c′(Λ)c(Λ)≥1e(Λ).\frac{c'(\Lambda)}{c(\Lambda)}\geq\frac{1}{e(\Lambda)}.

This asks whether the proportion of integers below the conductor that belong to the numerical semigroup is at least the reciprocal of its embedding dimension. The statement was posed by Wilf and has been supported by several authors; the source establishes only an asymptotic and approximate version, so the conjecture remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Wilf's conjecture for numerical semigroups

    Let SS be a numerical semigroup. Define

    ν(S)=∣{s∈S∣s≤F(S)}∣,\nu(S)=|\{s\in S\mid s\leq \mathrm{F}(S)\}|,

    where F(S)\mathrm{F}(S) is the Frobenius number of SS and e(S)\mathrm{e}(S) is the embedding dimension of SS. Wilf's conjecture. One has

    F(S)+1≤e(S)ν(S).\mathrm{F}(S)+1\leq \mathrm{e}(S)\mathrm{\nu}(S).

    Wilf's conjecture asserts a general bound relating the Frobenius number, embedding dimension, and the number of semigroup elements up to the Frobenius number. The candidate is presented as the statement of the conjecture; no resolution evidence is supplied here.

    source: Pranjal Srivastava and Dhara Thakkar, “The Frobenius Problem for the Proth Numbers”, arXiv:2311.12462 (2023).

Sources & referencesView supporting material

Primary source

Alex Zhai, “An asymptotic result concerning a question of Wilf”, arXiv:1111.2779 (2011).

combinatorial-number-theoryfrobenius-problemnumerical-semigroups

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