The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

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

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

The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].

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

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.