The skewness of a graph. For a graph $G$, the '''skewness''' of $G$ is the minimum number of edges of $G$ whose removal results in a planar graph.

The minimal crossing number of a graph. A '''drawing''' of a graph $G$ is a drawing in $\mathbb{R}^2$ such that * the vertices of $G$ are distinct points, * the edges of $G$ are simple curves joining their endpoints, * no edge passes through a vertex, and * no three edges cross in a common point. The '''minimal crossing number''' of $G$ is then the minimal number of crossings of edges in a drawing of $G$. In particular, a graph is planar if and only if its minimal crossing number is $0$. It is moreover conjectured that the crossing number of the complete graph $K_n$ [1] is $$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$$ and the crossing number of the complete bipartite graph $K_{n,m}$ [2] is $$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$$ A general algorithm to compute the crossing number is e.g. given in [3]. This statistics data was provided by Markus Chimani [6].

The number of factors DDU in a Dyck path.

The number of global maxima of a Dyck path.

The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

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