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In this article we solve the conjecture "Hilbert function of the local ringfor 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 acomplete characterization to the standard bases when the tangent cone is notCohen-Macaulay by showing that the number of elements in the standard basisdepends on some parameters $s_j$ 's we define. Since the tangent cone is notCohen-Macaulay, non-decreasingness of the Hilbert fuction was not guaranteed,we proved the non-decreasingness from our explicit Hilbert Functioncomputation.

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