Gapsets and numerical semigroups

Abstract

For g ≥ 0, let n g denote the number of numerical semigroups of genus g. A conjecture by Maria Bras-Amorós in 2008 states that the inequality n g ≥ n g−1 + n g−2 should hold for all g ≥ 2. Here we show that such an inequality holds for the very large subtree of numerical semigroups satisfying c ≤ 3m, where c and m are the conductor and multiplicity, respectively. Our proof is given in the more flexible setting of gapsets, i.e. complements in N of numerical semigroups.1

A Graph-Theoretic Approach to Wilf's Conjecture

“…Thus, from now on in this section, S is a numerical semigroup in multiplicative notation, arising from its additive counterpart S 0 ⊆ N via the isomorphism ϕ in (12). We denote S * = S \{1}.…”

“…Let S be a numerical semigroup in multiplicative notation, arising from a classical numerical semigroup S 0 ⊆ N via the isomorphism (12). The following notation will be used throughout Section 5.…”

Almost symmetric numerical semigroups with high type

“…Motivated by the notion of set of gaps of a numerical semigroup, in [9] the concept of gapset is introduced. Notice that if G is a gapset, then N \ G is a numerical semigroup.…”

“…We conclude Our correspondence provides a new characterization of almost symmetric numerical semigroups with high type. The depth of a numerical semigroup has shown to play a special role in the study of Wilf's conjecture (see for instance [6,9]). Let S be a numerical semigroup with Frobenius number F and multiplicity m, and write F + 1 = qm − r for some integers q and r with 0 ≤ r < m .…”

The right-generators descendant of a numerical semigroup

“…The nodes of this tree at level g are all numerical semigroups of genus g. The children, if any, of a given node arise through the removals of each of its right generators. The first levels are shown in Figure 1, where each semigroup is represented by its set of gaps, as in [9]. Proof.…”