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 Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$. Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The Frankl number of a lattice. For a lattice $L$ on at least two elements, this is $$ \max_x(|L|-2|[x, 1]|), $$ where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.

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.