The number of Coxeter elements in the Weyl group of a finite Cartan type. This is, the elements that are conjugate to the product of the simple generators in any order, or, equivalently, the elements that admit a primitive $h$-th root of unity as an eigenvalue where $h$ is the Coxeter number.

The largest coefficient in the Poincaré polynomial of the Weyl group of given Cartan type. The Poincaré polynomial of a Weyl group $W$ is $$ \sum_{w\in W} q^{\ell(w)} = \prod_i [d_i]_q, $$ where $\ell$ denotes the Coxeter length, $d_1,\dots$ are the degrees (or exponents) of $W$ and $[n]_q=1 +\dots+q^{n-1}$ is the $q$-integer. Thus, this statistic records the frequency of the most common length in the group.

The number of elements of maximal order in the Weyl group of a finite Cartan type. For the symmetric group $\mathfrak S_n$ this is [[OEIS:A074859]].

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.

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 maximum cut size of a graph. A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.

The Hosoya index of a graph. This is the total number of matchings in the graph.

The number of graphs with given frequency partition. The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur. For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$. There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.

The logarithm of the number of winning configurations of the lights out game on a graph. In the single player lamps out game, every vertex has two states, on or off. The player can toggle the state of a vertex, in which case all the neighbours of the vertex change state, too. The goal is to reach the configuration with all vertices off.