The Betti numbers and Golodness of numerical semigroup rings of Sally type

Abstract: We determine explicit formulae for the Betti numbers of the defining toric ideals of two families of numerical semigroups of Sally type, resolving and extending beyond several conjectures of Goel--Şahin--Singh--Srinivasan. Our approach uses Apéry resolutions and their specialisations. For the family [ Γ_m(n)=\langle m,m+1,\ldots,\widehat{m+n},\ldots,2m-1\rangle, \qquad 2\leq n<m, ] where the hat denotes omission, we also compute the Poincaré series of the residue field and characterise Golodness. For fixed multiplicity $m$ and $3\leq n&lt;m$, we show that these rings share the same rational residue field Poincaré series, but exhibit arbitrarily late departures from Serre's upper bound.