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

In this article we solve the conjecture "Hilbert function of the local ring for a 4 generated pseudo-symmetric numerical semigroup \(\langle n_1,n_2,n_3,n_4 \rangle\) is always non-decreasing when \( n_1 < n_2 < n_3 < n_4\)". We give a complete characterization to 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 \(s_j\) 's we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert fuction was not guaranteed, we proved the non-decreasingness from our explicit Hilbert Function computation.