The maximum down-degree of the Hasse diagram of the strong Bruhat order in the Weyl group of the Cartan type.

The number of types of parabolic subgroups of the associated Weyl group. Let $W$ be a Weyl group with simple generators $\mathcal{S} \subseteq W$. A subgroup of $W$ generated by a subset $X \subseteq \mathcal{S}$ is called standard parabolic subgroup. A parabolic subgroup is a subgroup of $W$ that is conjugate to a standard parabolic subgroup. This is the number of all pairwise different types of subgroups of $W$ obtained as (standard) parabolic subgroups (including type $A_0$).

The number of maximal chains in a poset.

The number of maximal chains of maximal size in a poset.

The number of maximal chains of minimal length in a poset.

The number of potential covers of a poset. A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations. For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.

The number of non-isomorphic posets with precisely one further covering relation.

The number of nonisomorphic vertex-induced subtrees.

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.

The number of orbits of vertices of a graph under automorphisms.