The number of subsets of the positive roots that form a basis of the associated vector space. For the group $W$ and an associated set of positive roots $\Phi^+ \subseteq V$ this counts the number of subsets $S \subseteq \Phi^+$ that form a basis of $V$. This is also the number of subsets of the reflections $R \subseteq W$ that form a minimal set of generators of a reflection subgroup of full rank. The Coxeter permutahedron can be defined as the Minkowski sum of the line segments $[- \frac{\alpha}{2}, \frac{\alpha}{2}]$ for $\alpha \in \Phi^+$. As a zonotope this polytope can be decomposed into a (disjoint) union of (half-open) parallel epipeds [1]. This also counts the number of full dimensional parallel epipeds among this decomposition.

The number of even entries in the character table of the Weyl group of a Cartan type.

The number of different adjacency matrices of a graph. This is the number of different labellings of the graph, or $\frac{|G|!}{|\operatorname{Aut}(G)|}$.

The Ramsey number of a graph. This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]

The number of linear intervals in a lattice.

The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The number of set partitions whose sorted block sizes correspond to the partition.

The number of permutations whose cycle type is the given integer partition. This number is given by $$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$$ where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$. All permutations with the same cycle type form a [[wikipedia:Conjugacy class]].

The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. Summing over all partitions of $n$ yields the sequence $$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$ which is [[oeis:A237770]]. The references in this sequence of the OEIS indicate a connection with Baxter permutations.