Near-misses in 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.

References in corpus (2)

• Fibonacci-Like Behavior of the Number of Numerical Semigroups of a Given Genus
• On a conjecture of Wilf about the Frobenius number

Cited by in corpus (7)

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• Trimming the numerical semigroups tree to probe Wilf's conjecture to higher genus
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• Iterated sumsets and Hilbert functions
• Eliahou number, Wilf function and concentration of a numerical semigroup