The sum of the vertex degrees of a graph. This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph. The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$

The number of potential covers of a poset. A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations. For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.

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 first Zagreb index of a graph. This is the sum of the squares of the degrees of the vertices, $$\sum_{v \in V(G)} d^2(v) = \sum_{\{u,v\}\in E(G)} \big(d(u)+d(v)\big)$$ where $d(u)$ is the degree of the vertex $u$.

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].

Twice the mean value of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by $$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$ where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column.

The dimension of the maximal parabolic seaweed algebra corresponding to the partition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be two compositions of $n$. The corresponding seaweed algebra is the associative subalgebra of the algebra of $n\times n$ matrices which preserves the flags $$ \{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V $$ and $$ V=W_0\supset W_1\supset \cdots \supset W_t=\{0\}, $$ where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$. Thus, its dimension is $$ \frac{1}{2}\left(\sum a_i^2 + \sum b_i^2\right). $$ It is maximal parabolic if $b_1=n$.

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 $\sum_{i=1}^n (\pi(i)−i)^2$.