The major index of a Dyck path. This is the sum over all $i+j$ for which $(i,j)$ is a valley of $D$. The generating function of the major index yields '''MacMahon''' 's $q$-Catalan numbers $$\sum_{D \in \mathfrak{D}_n} q^{\operatorname{maj}(D)} = \frac{1}{[n+1]_q}\begin{bmatrix} 2n \\ n \end{bmatrix}_q,$$ where $[k]_q := 1+q+\ldots+q^{k-1}$ is the usual $q$-extension of the integer $k$, $[k]_q!:= [1]_q[2]_q \cdots [k]_q$ is the $q$-factorial of $k$ and $\left[\begin{smallmatrix} k \\ l \end{smallmatrix}\right]_q:=[k]_q!/[l]_q![k-l]_q!$ is the $q$-binomial coefficient. The major index was first studied by P.A.MacMahon in [1], where he proved this generating function identity. There is a bijection $\psi$ between Dyck paths and '''noncrossing permutations''' which simultaneously sends the area of a Dyck path [[St000012]] to the number of inversions [[St000018]], and the major index of the Dyck path to $n(n-1)$ minus the sum of the major index and the major index of the inverse [2]. For the major index on other collections, see [[St000004]] for permutations and [[St000290]] for binary words.

The natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.

The comajor index of a standard tableau minus the weighted size of its shape.

The value of the forgotten symmetric functions when all variables set to 1. Let $f_\lambda(x)$ denote the forgotten symmetric functions. Then the statistic associated with $\lambda$, where $\lambda$ has $\ell$ parts, is $f_\lambda(1,1,\dotsc,1)$ where there are $\ell$ variables substituted by $1$.

Number of partial orders contained in the poset determined by the Dyck path. A Dyck path determines a poset, where the relations correspond to boxes under the path (seen as a North-East path). This statistic is closely related to unicellular LLT polynomials and their e-expansion.

Half of MacMahon's equal index of a Dyck path. This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].

Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path).

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

The number of parking functions supported by a Dyck path. One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.

The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.