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]].

Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

The number of natural descents for set-valued two row standard Young tableaux. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let $j$ be smallest integer such that $2j$ is a down step. Then $k$ is a natural descent if * $k-2\ge j$ and positions 2(k-1)-1,2(k-1) are a valley i.e. [0,1], or * $k-2\ge j$ and positions 2(k-1)-1,2(k-1) are a peak i.e. [1,0], or * $k-1\ge j$ and positions 2(k-1),2k-1,2k form [0,1,1], or * $k=j$ and positions 2k-1,2k are double down i.e. [0,0], or * $k < j$ and positions 2k-1,2k are a valley i.e. [0,1].

The number of centered multitunnels of a Dyck path. This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.