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.

The number of topologically connected components of a set partition. For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$. The number of set partitions with only one block is [[oeis:A099947]].

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

The number of critical left to right maxima of the parking functions. An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it. This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via $$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$ where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].