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 number of crossings plus two-nestings of a perfect matching. This is $C+2N$ where $C$ is the number of crossings ([[St000042]]) and $N$ is the number of nestings ([[St000041]]). The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)].

The sum of the descent tops (or Genocchi descents) of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.

The inversion index of a permutation. The ''inversion index'' of a permutation $\sigma=\sigma_1\sigma_2\ldots\sigma_n$ is defined as $$ \sum_{\mbox{inversion pairs } (\sigma_i,\sigma_j)} \sigma_i $$ where $(\sigma_i,\sigma_j)$ is an inversion pair if i < j and $\sigma_i > \sigma_j$. This equals the sum of the entries in the corresponding descending plane partition; see [1].

The sum of the major and the inverse major index of a permutation. This statistic is the sum of [[St000004]] and [[St000305]].

The spearman's rho of a permutation and the identity permutation. This is, for a permutation $\pi$ of $n$, given by $\sum_{i=1}^n (\pi(i)−i)^2$.

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].

The projective dimension of the second term in a minimal injective coresolution of the regular module.

The number of inversions plus the major index of a permutation. This is, the sum of [[St000004]] and [[St000018]].