theta functions $\phi_1, \phi_0, \psi_1, \psi_0$ in terms of Dyck paths and their comajor index (see Corollary \[mockcoef\]).

-   We find a combinatorial interpretation of the quotients of powers of two Chebyshev polynomials in terms of Dyck paths of bounded height generalizing the results of Gessel and Xin [@GX05] (see Proposition \[numerical\]).

-   The powers of $q$ which appear with non–zero coefficients in the polynomial encoding the graded multiplicities in Demazure flags of local Weyl modules form an interval of consecutive integers (see Corollary \[conint\]).

-   The graded multiplicities of irreducible modules in local Weyl modules are principal specializations of Schur functions (see Corollary \[princp\]).

-   We give formulae for the graded multiplicities of an irreducible module in a Demazure flag of a local Weyl module (see Corollary \[KrM\]).

Our paper is organized as follows. In Section 2 we recall the notion of excellent filtrations. In Section 3 we define our combinatorial model which is the crucial object in our main theorem. In Section 4 we reduce the computation of graded multiplicities in excellent filtrations of fusion products to the computation of graded multiplicities in local Weyl modules and state our main theorem. We also discuss several corollaries. In Section 