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

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

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 inner corners of the parallelogram polyomino associated with the Dyck path.

The number of maximal chains in a poset.

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 (zero)-forcing number of a graph. This is the minimal number of vertices initially coloured black, such that eventually all vertices of the graph are coloured black when using the following rule: when $u$ is a black vertex of $G$, and exactly one neighbour $v$ of $u$ is white, then colour $v$ black.

The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

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

The number of ones in a binary word. This is also known as the Hamming weight of the word.