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

Manuel Delgado, Neeraj Kumar, Claude Marion

TL;DR

This paper introduces counting numerical semigroups by maximum primitive, defining $A_n$ and relating it to the Frobenius-based counts $N_n$ via Möbius transforms. It proves $A_n$ and $N_n$ are asymptotically equivalent and proves almost all semigroups with large maximum primitive satisfy Wilf's conjecture, building on a key criterion: if a semigroup has many left-over elements in $(m,2m)$ then Wilf holds. A central tool is the injective but not surjective map $\Phi$ between the max-primitive and Frobenius-counting families, plus a divisor-based partition linking $A_n$ and $N_n$ through $N_n = \sum_{d|n} A_d$ and Möbius inversion. The results yield explicit growth bounds, lower bounds, and asymptotics for $A_n$, and establish that almost all $S\in\mathcal{A}_n$ satisfy Wilf as $n\to\infty$, with new infinite Wilf-satisfying classes arising from the derived criterion. Overall, the work deepens the connection between different counting paradigms in numerical semigroups and extends Wilf-type results to a broader combinatorial setting.

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. For any positive integer $n$, let $A_{n}$ denote the number of numerical semigroups whose maximum primitive is $n$, and let $N_{n}$ denote the number of numerical semigroups whose Frobenius number is $n$. We show that the sequences $(A_{n})$ and $(N_{n})$ are Möbius 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 $|S\cap (\mathrm{m},2 \mathrm{m})|\geq \sqrt{3\mathrm{m}}$ satisfies Wilf's conjecture.

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

TL;DR

This paper introduces counting numerical semigroups by maximum primitive, defining and relating it to the Frobenius-based counts via Möbius transforms. It proves and are asymptotically equivalent and proves almost all semigroups with large maximum primitive satisfy Wilf's conjecture, building on a key criterion: if a semigroup has many left-over elements in then Wilf holds. A central tool is the injective but not surjective map between the max-primitive and Frobenius-counting families, plus a divisor-based partition linking and through and Möbius inversion. The results yield explicit growth bounds, lower bounds, and asymptotics for , and establish that almost all satisfy Wilf as , with new infinite Wilf-satisfying classes arising from the derived criterion. Overall, the work deepens the connection between different counting paradigms in numerical semigroups and extends Wilf-type results to a broader combinatorial setting.

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. For any positive integer , let denote the number of numerical semigroups whose maximum primitive is , and let denote the number of numerical semigroups whose Frobenius number is . We show that the sequences and are Möbius 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 with multiplicity such that satisfies Wilf's conjecture.
Paper Structure (9 sections, 43 theorems, 100 equations, 3 figures, 1 table)

This paper contains 9 sections, 43 theorems, 100 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The map $\Phi$
  4. A partition of $\mathcal{N}_f$
  5. Asymptotic growth of $A_f$
  6. Bounds on $A_f$
  7. A correspondence between both sequences
  8. On Wilf's Conjecture
  9. Concluding remarks

Key Result

Proposition 1.1

For every $n>0$, ${A}_{n}<{N}_{n}$. Moreover, for a prime $p$, ${A}_{p}={N}_{p}-1$.

Figures (3)

  • Figure 2.1: Pictorial views of $S=\langle 6,7,8\rangle$ and $T=\langle 10,11,\ldots,19\rangle$.
  • Figure 3.1: Pictorial views of $S=\langle 5,6\rangle$ and $\Phi(S)=\langle 5,7,8,9,11\rangle$.
  • Figure 3.2: Pictorial views of $S=\langle 10,\ldots,19\rangle$ and $\Phi(S)=\langle 10,\ldots,18\rangle$.

Theorems & Definitions (73)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 63 more