Y-proper graded cocharacters of upper triangular matrices of order m graded by the m-tuple ϕ=(0, 0, 1,..., m-2)

Let F be a field of characteristic 0. We consider the algebra UTm(F) of upper triangular matrices of order m endowed with an elementary Zm-grading induced by the m-tuple ϕ=(0, 0, 1,..., m-2), then we compute its Y-proper graded cocharacter sequence and we give the explicit formulas for the multiplicities in the case m=2, 3, 4, 5