On numerical semigroups with at most 12 left elements

TL;DR

This paper proves Wilf's conjecture holds for numerical semigroups with at most 12 left elements, using a new lower bound, and shows the bound is tight by providing counterexamples at 13 left elements.

Contribution

The paper establishes Wilf's conjecture for all numerical semigroups with up to 12 left elements, introducing a new lower bound that confirms the conjecture in this range.

Findings

Wilf's conjecture is true for |L| ≤ 12.

A new lower bound W0(S) ≥ 0 is introduced and used.

Counterexamples exist for |L| = 13, where W0(S) = -1.

Abstract

For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) $\le$ W (S), we show that if |L| $\le$ 12 then W 0 (S) $\ge$ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilf's conjecture remains open for |L| $\ge$ 13.