Y-proper graded cocharacters and codimensions of upper triangular matrices of size 2, 3, 4

Let F be a field of characteristic 0. We consider the upper triangular matrices with entries in F of size 2, 3 and 4 endowed with the grading induced by that of Vasilovsky. In this paper we give explicit computation for the multiplicities of the Y-proper graded cocharacters and codimensions of these algebras.