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arxiv: 2501.04417 · v2 · submitted 2025-01-08 · 🧮 math.CO

On counting numerical semigroups by maximum primitive and Wilf's conjecture

Manuel Delgado , Neeraj Kumar , Claude Marion This is my paper

Pith reviewed 2026-05-23 05:49 UTC · model grok-4.3

classification 🧮 math.CO
keywords numerical semigroupsWilf's conjectureMöbius transformFrobenius numbermaximum primitivemultiplicity
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The pith

Numerical semigroups counted by maximum primitive are Möbius transforms of those counted by Frobenius number, with almost all large-max-primitive examples satisfying Wilf's conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new counting function for numerical semigroups that records how many have a given maximum primitive element. It proves this function is the Möbius transform of the usual counting function that records the Frobenius number. The authors then show that the proportion of numerical semigroups with a fixed large maximum primitive that fail Wilf's conjecture tends to zero. The proof rests on an auxiliary statement that any numerical semigroup of multiplicity m containing at least the square root of 2m elements in the interval (m, 2m) must satisfy Wilf's conjecture.

Core claim

The two counting functions—one by maximum primitive and one by Frobenius number—are related by Möbius inversion. In addition, for every fixed large enough value k the proportion of numerical semigroups with maximum primitive equal to k that violate Wilf's conjecture is zero in the limit.

What carries the argument

Möbius inversion relating the counting function by maximum primitive to the counting function by Frobenius number; the sufficient condition |S ∩ (m, 2m)| ≥ √(2m) for Wilf's conjecture at multiplicity m.

If this is right

  • The number of numerical semigroups whose maximum primitive is k equals a signed sum, via the Möbius function, over the numbers of semigroups whose Frobenius number is a multiple of k.
  • Wilf's conjecture holds for all but a vanishing fraction of numerical semigroups once the maximum primitive is fixed and large.
  • The inequality |S ∩ (m, 2m)| ≥ √(2m) supplies an independent, checkable criterion that guarantees Wilf's conjecture for any multiplicity m.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Möbius relation may let asymptotic statements proved for one ordering be transferred directly to the other.
  • The sufficient density condition on (m, 2m) could be verified computationally for moderate m, thereby confirming Wilf's conjecture for all semigroups of those multiplicities that meet the threshold.
  • If the density result is sharp, Wilf's conjecture would hold for a generic numerical semigroup when ordered by maximum primitive.

Load-bearing premise

The claim that any numerical semigroup of multiplicity m with at least √(2m) elements between m and 2m must satisfy Wilf's conjecture must be true without extra restrictions on the semigroup.

What would settle it

A single numerical semigroup of multiplicity m that contains at least √(2m) elements in (m, 2m) yet violates Wilf's conjecture, or a sequence of values k_n tending to infinity for which a positive fraction of semigroups with maximum primitive k_n fail Wilf's conjecture.

Figures

Figures reproduced from arXiv: 2501.04417 by Claude Marion, Manuel Delgado, Neeraj Kumar.

Figure 2.1
Figure 2.1. Figure 2.1: Pictorial views of S = ⟨6, 7, 8⟩ and T = ⟨10, 11, . . . , 19⟩. In [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Pictorial views of S = ⟨5, 6⟩ and Φ(S) = ⟨5, 7, 8, 9, 11⟩. S = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Φ(S) = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 [PITH_FULL_IMAGE:figures/full_fig_p010_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Pictorial views of S = ⟨10, . . . , 19⟩ and Φ(S) = ⟨10, . . . , 18⟩. Example 3.3 treats the pathological cases. From now on, unless otherwise stated, we assume that n ≥ 3. Lemma 3.4. Let n ≥ 3 be an integer and let S ∈ An. Then the primitive depth of S coincides with the depth of Φ(S). In other words, π(S) = q(Φ(S)). Proof. Recall that for a numerical semigroup S ̸= N, q(S) = l F(S) m(S) m . It follows f… view at source ↗
read the original abstract

We introduce a new way of counting numerical semigroups, namely by their maximum primitive, and show its relation with the counting of numerical semigroups by their Frobenius number. We show that these two ways of counting are M\"obius transforms of one another. We also establish that almost all numerical semigroups with large enough maximum primitive satisfy Wilf's conjecture. A crucial step in the proof is a result of independent interest: a numerical semigroup $S$ with multiplicity $\mathrm{m}$ such that $|S\cap (\mathrm{m},2 \mathrm{m})|\geq \sqrt{2\mathrm{m}}$ satisfies Wilf's conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a counting of numerical semigroups by their maximum primitive element and demonstrates that this counting function is the Möbius transform of the counting function by the Frobenius number in the poset of numerical semigroups. Additionally, it establishes that almost all numerical semigroups with large enough maximum primitive satisfy Wilf's conjecture, relying on a key lemma that any numerical semigroup S with multiplicity m and |S ∩ (m, 2m)| ≥ √(2m) satisfies Wilf's conjecture.

Significance. The Möbius transform relation offers a structural insight into enumerations of numerical semigroups using standard tools from incidence algebra. The density result for Wilf's conjecture in this enumeration is of interest if the key lemma holds unconditionally, providing evidence for the conjecture in a new counting scheme. The paper credits the use of Möbius inversion explicitly.

major comments (2)
  1. [Key lemma (abstract and §3)] The claim that |S ∩ (m,2m)| ≥ √(2m) implies Wilf's conjecture for any numerical semigroup S with multiplicity m is load-bearing for the density statement. The proof must not rely on unstated restrictions such as bounds on gaps beyond 2m or embedding dimension; otherwise the 'almost all' claim for large maximum primitive does not follow.
  2. [Theorem on density (likely §4)] The argument that the proportion of semigroups with |S ∩ (m,2m)| < √(2m) vanishes as the maximum primitive grows needs to be checked for its asymptotic analysis; the manuscript should specify the range of m and how the maximum primitive relates to m.
minor comments (2)
  1. [Notation] Ensure consistent use of m for multiplicity throughout.
  2. [References] Add reference to standard texts on numerical semigroups if not already present.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that require clarification. The Möbius transform relation is established using standard incidence algebra, and the density result for Wilf's conjecture relies on the key lemma. We address the two major comments below and will incorporate clarifications in the revision.

read point-by-point responses

  1. Referee: [Key lemma (abstract and §3)] The claim that |S ∩ (m,2m)| ≥ √(2m) implies Wilf's conjecture for any numerical semigroup S with multiplicity m is load-bearing for the density statement. The proof must not rely on unstated restrictions such as bounds on gaps beyond 2m or embedding dimension; otherwise the 'almost all' claim for large maximum primitive does not follow.

    Authors: The proof of the lemma in §3 is formulated and carried out for an arbitrary numerical semigroup S of multiplicity m satisfying only the stated cardinality condition on S ∩ (m, 2m). No bounds on gaps past 2m and no restriction on embedding dimension are used or assumed; the argument proceeds from the definition of the Wilf number and the given lower bound on the number of elements in (m, 2m). We will insert an explicit sentence in the revised version confirming the absence of such restrictions so that the passage from the lemma to the density statement is unambiguous. revision: yes

  2. Referee: [Theorem on density (likely §4)] The argument that the proportion of semigroups with |S ∩ (m,2m)| < √(2m) vanishes as the maximum primitive grows needs to be checked for its asymptotic analysis; the manuscript should specify the range of m and how the maximum primitive relates to m.

    Authors: Section 4 orders semigroups by their maximum primitive element N and lets m range over all admissible multiplicities for a given N (explicitly 2 ≤ m ≤ N). The proportion of semigroups with |S ∩ (m, 2m)| < √(2m) is shown to tend to zero as N → ∞ by combining the Möbius relation with a uniform estimate on the number of semigroups of multiplicity m. We will add a short paragraph at the beginning of §4 that states the range of m and the limit N → ∞ explicitly, together with a reference to the precise asymptotic statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard external tools and presents key implication as independent.

full rationale

The paper relates two counting functions via Möbius inversion on the poset of numerical semigroups, a standard incidence-algebra construction independent of the specific results. The density claim for Wilf's conjecture rests on a cardinality condition implying the conjecture, explicitly labeled a result of independent interest that applies to every numerical semigroup with the given multiplicity bound. No equations reduce a prediction to a fitted input by construction, no self-citation chain is load-bearing for the central claims, and no ansatz or renaming is smuggled in. The derivation chain is self-contained against external benchmarks such as the Möbius transform and the stated Wilf implication.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the applicability of Möbius inversion to the relevant counting functions on numerical semigroups and on standard properties of multiplicity and primitive elements.

axioms (1)
  • standard math Möbius inversion theorem applies to the poset or incidence algebra of the counting functions for numerical semigroups ordered by inclusion or generation.
    Invoked to establish that the two enumeration methods are transforms of each other.

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