Near-misses in Wilf's conjecture

TL;DR

This paper investigates near-misses to Wilf's conjecture in numerical semigroups, constructing infinite families of near-misses with specific properties, and demonstrates that these near-misses still satisfy the conjecture.

Contribution

The paper introduces the concept of near-misses in Wilf's conjecture, constructs infinite families of such near-misses with particular parameters, and shows they do not violate the conjecture.

Findings

Infinite families of near-misses with arbitrarily small W0(S) are constructed.

Members of these families satisfy Wilf's conjecture despite being near-misses.

Near-misses are shown to be very rare in the context of Wilf's conjecture.

Abstract

Let S $\subseteq$ N be a numerical semigroup with multiplicity m, conductor c and minimal generating set P. Let L = S $\cap$ [0, c -- 1] and W(S) = |P||L| -- c. In 1978, Herbert Wilf asked whether W(S) $\ge$ 0 always holds, a question known as Wilf's conjecture and open since then. A related number W0(S), satisfying W0(S) $\le$ W(S), has recently been introduced. We say that S is a near-miss in Wilf's conjecture if W0(S) < 0. Near-misses are very rare. Here we construct infinite families of them, with c = 4m and W0(S) arbitrarily small, and we show that the members of these families still satisfy Wilf's conjecture.