major role in the description of the generating series and graded multiplicities respectively, see Theorem \[mainthm1\].

We have $A(n,n)=\{(0,n)\}$ and hence $\mathcal{D}_{n,n}$ consists of all Dyck paths $P\in \mathcal{D}^n_{n}$ which do not return to the $x$–axis after a peak of height $n$. The green path in Figure \[fig3\] is admissible and the red path violates the condition and is not admissible.

(0,0) grid (16,5); (0,0) – (4,4) – (6,2) – (7,3) – (8,2) – (10,4) – (13,1) – (16,4);

(0,0) grid (16,5); (0,0) – (4,4) – (6,2) – (7,3) – (10,0) – (12,2) – (13,1) – (16,4);

 {#section35}

The major statistics of a Dyck path was studied first by MacMahon [@MacM60] in his interpretation of the $q$–Catalan numbers. Let $P=a_1\cdots a_{s}$, $a_i\in\{0,1\}$ a Dyck path of length $s$. The major and comajor index are defined by $$\text{maj}(P)=\sum_{\substack{1\leq i < s,\\ a_i>a_{i+1}}} i,\ \ \ \text{comaj}(P)=\sum_{\substack{1\leq i < s,\\ a_i>a_{i+1}}} (s-i).$$

\[rem1\] The comajor index can also be defined for standard Young tableaux and play an important role in the expression of the character of the space of harmonics (the vector space spanned by the Vandermonde determinant and its partial derivatives of all orders; see the work of Haiman [@Hai94]). The 