Density of \(f\)-ideals and \(f\)-ideals in mixed small degrees

A squarefree monomial ideal is called an \(f\)-ideal if its Stanley-Reisner and facet simplicial complexes have the same \(f\)-vector. We show that \(f\)-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct \(f\)-ideals generated in small degrees.