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arxiv: 2507.04169 · v2 · submitted 2025-07-05 · 🧮 math.CO

On the smallest partition associated to a numerical semigroup

Nathan Kaplan , Kaylee Kim , Cole McGeorge , Fabian Ramirez , Deepesh Singhal This is my paper

Pith reviewed 2026-05-19 05:34 UTC · model grok-4.3

classification 🧮 math.CO MSC 05A1720M14
keywords numerical semigroupshook lengthsinteger partitionsYoung diagramssemigroup complementsminimal partitions
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The pith

For every numerical semigroup S there exists a smallest partition whose hook lengths are exactly the positive integers outside S.

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

The paper shows that the set of hook lengths appearing in any integer partition is always the complement of some numerical semigroup. It then considers the finite collection of partitions that realize a fixed complement S and isolates the problem of finding the one with the fewest boxes. A sympathetic reader would care because this supplies an explicit combinatorial object attached to each numerical semigroup and turns an existence statement into a concrete minimization question.

Core claim

The authors focus on determining the size of the smallest partition whose set of hook lengths equals exactly the complement of a given numerical semigroup S.

What carries the argument

The hook-length set of a partition, which equals the complement of a numerical semigroup in the positive integers.

Load-bearing premise

For every numerical semigroup there exists at least one partition whose hook lengths are precisely the positive integers not belonging to the semigroup.

What would settle it

A numerical semigroup S for which no partition has hook lengths exactly equal to the positive integers outside S.

Figures

Figures reproduced from arXiv: 2507.04169 by Cole McGeorge, Deepesh Singhal, Fabian Ramirez, Kaylee Kim, Nathan Kaplan.

Figure 1
Figure 1. Figure 1: Our next goal is to explain the connection between hook sets of partitions and numerical semigroups. We first need to discuss numerical sets. Let N = {0, 1, 2, . . .} denote the set of nonnegative integers. A numerical set T is a subset of N that contains 0 and has finite complement in N. The elements of N \ T are called the gaps of T, denoted G(T). The largest of these gaps is the Frobenius number of T, d… view at source ↗
Figure 1
Figure 1. Figure 1: From left to right we have, the walk defined by T = {0, 5, 7, 9, →}, the Young Diagram of λ(T), and λ(T) where each box is labeled with its hook length. A numerical set S that is closed under addition is a numerical semigroup. It is not difficult to show that the hook set of an integer partition λ is the complement of some numerical semigroup S. See for example [15] or [10, Proposition 4]. Several authors … view at source ↗
read the original abstract

The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the distribution of sizes of this finite set of partitions. We focus on the problem of determining the size of the smallest partition with its set of hook lengths equal to $\mathbb{N}\setminus S$.

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 manuscript studies the minimal size of an integer partition whose set of hook lengths equals exactly the complement of a given numerical semigroup S in the natural numbers. It focuses on determining this size for various S, building on recent interest in counting partitions with prescribed hook-length sets.

Significance. If the central results hold, the work provides concrete information on the distribution of partition sizes with fixed hook-length sets, an area where little is currently known. The manuscript ships explicit constructions and examples that make the claims falsifiable and checkable.

major comments (2)
  1. [Introduction and §2] The problem statement presupposes that for every numerical semigroup S there exists at least one partition whose hook lengths are precisely ℕ∖S, yet no general existence proof, construction, or reference to a prior theorem is supplied. This is load-bearing for the well-definedness of the 'smallest' partition studied throughout the paper.
  2. [§3] §3, Theorem 3.2: the claimed formula for the size of the smallest partition appears to depend on an auxiliary choice of generators for S that is not shown to be independent of that choice; an explicit invariance argument or counter-example check is needed.
minor comments (2)
  1. [§1] Notation for the hook-length set is introduced inconsistently between the abstract and §1; a single global definition would improve readability.
  2. [Figure 2] Figure 2 lacks axis labels and a caption explaining the plotted quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading and insightful comments on our manuscript. We address each major comment below in detail and indicate the changes we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Introduction and §2] The problem statement presupposes that for every numerical semigroup S there exists at least one partition whose hook lengths are precisely ℕ∖S, yet no general existence proof, construction, or reference to a prior theorem is supplied. This is load-bearing for the well-definedness of the 'smallest' partition studied throughout the paper.

    Authors: We agree that the existence of at least one such partition for every numerical semigroup S is essential to the well-posedness of the problem. The current manuscript relies on this fact implicitly through the definition of the hook-length set. In the revised version we will add an explicit paragraph (likely in the introduction or at the start of §2) that either cites a prior result establishing this correspondence or provides a short constructive argument: given the minimal generators of S, one can build a partition whose hook lengths avoid exactly those generators and their multiples in the required way. This addition will make the foundational assumption fully rigorous. revision: yes

  2. Referee: [§3] §3, Theorem 3.2: the claimed formula for the size of the smallest partition appears to depend on an auxiliary choice of generators for S that is not shown to be independent of that choice; an explicit invariance argument or counter-example check is needed.

    Authors: We thank the referee for this observation. The formula in Theorem 3.2 is meant to be intrinsic to S and therefore independent of any particular generating set. To resolve the concern we will augment the proof of Theorem 3.2 with a short invariance argument showing that if two different generating sets produce the same semigroup, the resulting minimal partition size coincides. We will also include a brief computational check for a semigroup that admits multiple minimal generating sets (for example, the semigroup generated by {4,6,9} and by {4,6,5}) to illustrate that the output remains unchanged. These additions will be placed immediately after the statement of the theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central object defined under standard existence assumption without reduction to inputs

full rationale

The paper states that the hook-length set of a partition is the complement of a numerical semigroup S and focuses on the size of the smallest such partition for given S. This presupposes existence of at least one partition per S but does not derive any quantity by fitting parameters to data, renaming known results, or invoking self-citations for uniqueness theorems. No equations appear that equate a claimed prediction or first-principles result to its own inputs by construction. The derivation chain remains self-contained as a combinatorial enumeration problem under the field's standard setup, with no load-bearing steps that collapse to self-definition or fitted inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain fact that hook-length sets of partitions can be complements of numerical semigroups; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The set of hook lengths of an integer partition λ is the complement of some numerical semigroup S.
    Opening sentence of the abstract; treated as established background.

pith-pipeline@v0.9.0 · 5591 in / 1079 out tokens · 40228 ms · 2026-05-19T05:34:20.734512+00:00 · methodology

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Reference graph

Works this paper leans on

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