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].

Number of simple reflexive modules that are 2-stable reflexive. See Definition 3.1. in the reference for the definition of 2-stable reflexive.

The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations.