The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

The sum of the parts of an integer partition that are at least two.

The rank of the adjacency matrix of a graph.

The proper pathwidth of a graph. The proper pathwidth $\operatorname{ppw}(G)$ was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if $G$ has at least one edge, then $\operatorname{ppw}(G)$ is the minimum $k$ for which $G$ is a minor of the Cartesian product $K_k \square P$ of a complete graph on $k$ vertices with a path; and further that $\operatorname{ppw}(G)$ is the minor monotone floor $\lfloor \operatorname{Z} \rfloor(G) := \min\{\operatorname{Z}(H) \mid G \preceq H\}$ of the [[St000482|zero forcing number]] $\operatorname{Z}(G)$. It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for $H$ in this definition, i.e. $\lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}$. The minimum degree $\delta$, treewidth $\operatorname{tw}$, and pathwidth $\operatorname{pw}$ satisfy $$\delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1.$$ Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.

The number of distinct eigenvalues of a graph.

The sum of the positions of the strong records of an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. This statistic is the sum of the positions of the strong records.

The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.