The number of maximal chains in a poset.

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

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.

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.

The number of maximally independent sets of vertices of a graph. An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.

The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.

The tree-depth of a graph. The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as $$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$ Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once). Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].