On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: "The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing." We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.