The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.

The number of pairs of vertices of a graph with distance 3. This is the coefficient of the cubic term of the Wiener polynomial, also called Wiener polarity index.

The number of induced paths on four vertices in a graph.

The minimal number of edges to add or remove to make a graph a cograph. A cograph is a graph that can be obtained from the one vertex graph by complementation and disjoint union.

The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph.

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Half of the Albertson index of a graph. This is $\frac{1}{2}\sum_{\{u,v\}\in E}|d(u)-d(v)|$, where $E$ is the set of edges and $d_v$ is the degree of vertex $v$, see [1]. In particular, this statistic vanishes on graphs whose components are all regular, see [2].

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

The minimal number of edges to add or remove to make a graph edge transitive. A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.