The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].

The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [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 topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$. This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.