Let \(F\) be a field of characteristic 0, and let \(A=UT_m(F)\) be the associative algebra of the upper triangular \(m\times m\) matrices over \(F\). Suppose \(G\) is a finite group and that \(A\) is \(G\)-graded. It was proved by A. Valenti and M. V. Zaicev [Arch. Math. 89, No. 1, 33–40 (2007; Zbl 1151.16042)] that every group grading on \(A\) is isomorphic to elementary one; the elementary gradings on \(A\) and the corresponding graded identities were described by O. M. Di Vincenzo et al. [J. Algebra 275, No. 2, 550–566 (2004; Zbl 1066.16047)]. Since the base field is of characteristic 0 one studies the multilinear polynomial identities. The numerical characterization of the ideals of identities is an important part of the theory of PI algebras. Among the most prominent numerical invariants of an ideal of identities is its cocharacter sequence. Recall that the vector space \(P_n\) of all multilinear elements of degree \(n\) in the free associative algebra is a left module over the symmetric group \(S_n\) by permuting the variables \(x_1, \dots, x_n\). If \(I=Id(R)\) is the T-ideal of an algebra \(R\) then \(I\) is clearly \(S_n\)-invariant, and \(P_n\cap I\) is a submodule. One studies the quotient \(P_n/(P_n\cap I) = P_n(R)\) as an \(S_n\)-module. Its character is the \(n\)-th cocharacter of \(R\). If \(R\) is \(G\)-graded one modifies accordingly the above notions and one has the \(n\)-th graded cocharacter of \(R\). In the case of ordinary identities of unitary algebras one can consider the so-called proper multilinear elements: the multilinear polynomials that can be written as linear combinations of (long) commutators. These also generate the corresponding T-ideal, and furthermore form a smaller set. The analogous notion in the graded case is represented by the \(Y\)-proper elements. These are defined as follows. Let the set of the free generators of the free associative algebra be \(X\). Split \(X\) as a disjoint union of the infinite sets \(X_g\), \(g\in G\), and grade the free algebra \(F\langle X\rangle\) by assuming the variables of \(X_g\) are of degree \(g\). Then set \(Y=X_1\), \(Z=\cup_{g\ne 1} X_g\). Since the unit element of \(R\) lies in the neutral component of the grading on \(R\) one can restrict the class of polynomials to the \(Y\)-proper ones: these where each \(Y\)-variable always appears inside commutators. The authors describe the \(Y\)-proper cocharacters of \(A=UT_m(F)\) for an arbitrary (elementary) grading by a finite group. The authors also relate the information about the proper cocharacters to the growth of the corresponding ideal of graded identities, and to the Gelfand-Kirillov dimension in the graded case.