The stabilised diagonals of n(g,t) and ideals of numerical semigroups

Abstract

Let n(g,t) be the number of numerical semigroups of genus g and type t. Chappelon, Ramírez Alfonsín and Stamate proved that each diagonal t=g-ℓ of this array is eventually constant, n(g,g-ℓ)=|V(ℓ)| for g ≥ 3ℓ-1, where V(ℓ) is a set of balanced binary words of length 2ℓ prescribed by a condition on their difference set; they tabulated |V(ℓ)| for ℓ ≤ 11, asked for |V(ℓ)| in general, and proved only |V(ℓ)| ≥ ℓ²-3ℓ+10 — which at ℓ=11 reads 98 against the true value 3436. We show that the defining condition is a statement about numerical semigroups after all: the cotype of a word is the genus of the multiplier semigroup of an associated ideal, so that V(ℓ) is the set of subsets I ⊆ ℕ with ℓ gaps whose multiplier semigroup has genus ℓ. From the largest fibre of this description we deduce |V(ℓ)| ≥ 2^(k(⌊ ℓ/(k+1)⌋-1)) for every k ≥ 1, hence |V(ℓ)| ≥ 2^(ℓ-2√(ℓ)+O(1)), and a sharper union bound 2^(ℓ-log₂ℓ-O(1)); the same fibre turns out to be a sequence counted by Marzuola and Miller in 2010, in a paper on a different indexing of numerical sets, which upgrades the answer to |V(ℓ)| ≥ γ_∞ · 2^(ℓ-1) for their density constant γ_∞=0.4844 ± 0.0051. We also compute |V(12)|=7234, the first value beyond the published table, so that n(g,g-12)=7234 for every g ≥ 35, and prove n(35,23) ≥ 7234 from explicit numerical semigroups; since the published table gives n(33,21)=7214, the value 7234 is not recoverable from it. All statements are machine-checked in Lean 4.

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