Algorithms for numerical semigroups with fixed maximum primitive

TL;DR

The paper tackles counting numerical semigroups with a fixed maximum primitive $M$ and verifies Wilf's conjecture within this finite family. It develops both naive and optimized algorithms, including a Möbius-inversion based counting approach and pruning strategies (e.g., PossibleLargePrimitives) alongside a tree-based enumeration in GAP, to enable efficient computation up to $M\le 60$. A key theoretical contribution is the formula for depth-2 counts, $A_n(2)=\sum_{d|n}\mu(n/d)(2^{\lfloor (d-1)/2\rfloor}-1)$, derived via a depth-parameter bijection and Möbius inversion; this informs counting without exhaustive enumeration. Computational results yield new exact values $A_{61}=2640706082$, $A_{62}=2606696049$, and $N_{61}=2640706083$, $N_{62}=2606766903$, and confirm Wilf's conjecture for all $\mathcal{A}_n$ with $n\le 60$, contributing new data and supporting the conjecture within a substantial finite class while highlighting new examples beyond known sufficient conditions.

Abstract

We present an algorithm to explore various properties of the numerical semigroups with a given maximum primitive. In particular, we count the number of such numerical semigroups and verify that there is no counterexample to Wilf's conjecture among the numerical semigroups with maximum primitive up to \(60\).

Figures (2)

• Figure 1: ${A}_{M}$ using logarithmic scale in the $y$-axis
• Figure 2: The behavior of the sequences of ratios ${A}_{2n}/{A}_{2n+1}$ and ${A}_{2n+1}/{A}_{2n}$

Theorems & Definitions (17)

• Proposition 1
• Proposition 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• Lemma 8
• Proposition 9