flag of an arbitrary fusion product and to further understand the deep and unexpected link between the theory of Demazure flags and combinatorics. The main ingredients in our study are two dimensional lattice paths.

A lattice path is a path in a lattice in some euclidean vector space. They have a long history and have entered many fields of mathematic, computer science and physics; for a survey of results in the enumeration of lattice paths we refer the reader to [@Kra15]. A diagonal lattice path which consist only of up–steps $(1,1)$ and down–steps $(1,-1)$ and which do not pass below the $x$–axis is called a Dyck path. There are several statistics on the set of Dyck paths and we are interested in the comajor statistics (see Section \[section35\]) which was studied first by MacMahon [@MacM60] in his interpretation of the $q$–Catalan numbers. Our strategy for determining graded multiplicities in excellent filtrations of fusion products is as follows.

Let $\xi$ a partition and $V(\xi)$ the fusion product associated to $\xi$. We fix a level $m$–Demazure module $\tau_p^* D(m,n)$ of heighest weight $n$ whose degree is $p$ (see Section \[section2\] for the precise definitions). Further, set $$\mathcal{V}_{n}^{\xi\rightarrow m}(q)=\sum_{p\ge 0}\ [V(\xi):\tau_p^* D(m,n)] \hspace{0,04cm} 