The number of up-down runs of a permutation. An '''up-down run''' of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$. For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and $643$.

The number of distinct Laplacian eigenvalues of a graph.

The number of orbits of vertices of a graph under automorphisms.

The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.

The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

The number of internal nodes in the modular decomposition of a graph.

The stat' of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.