The number of indices that are both descents and recoils of a permutation.

The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.

The excess length of a longest path consisting of elements and blocks of a set partition. Let $p$ be a set partition of $\{1,\dots,n\}$. Let $G$ be the graph with edges $(i,i+1)$ for $i\in\{1,\dots,n-1\}$ and $(i, b)$, whenever $i$ is an element of a non-singleton block $b\in p$. Then this statistic records the length of the longest path from $1$ to $n$ in $G$, reduced by $n$. Conjecturally, a longest path has more than $n$ vertices provided that the set partition has no singletons.

The number of adjacent transpositions in the cycle decomposition of a permutation.

The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.

The number of adjacencies of a permutation. An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.

The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.