The number of attacking pairs of a standard tableau. Note that this is actually a statistic on the underlying partition. A pair of cells $(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds: 1. $c$ and $d$ lie in the same row with $c$ strictly to the west of $d$. 2. $c$ is in the row immediately to the south of $d$, and $c$ lies strictly east of $d$.

The dinv of a Dyck path. Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]). The dinv statistic of $D$ is $$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$ Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by $$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$

The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.