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

The number of linear intervals in a lattice.

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.

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$, where $k$ is the number of parts of $\lambda$. An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.

The Kreweras number of an integer partition. This is defined for $\lambda \vdash n$ with $k$ parts as $$\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}$$ where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition $\lambda$. These numbers refine the Narayana numbers $N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}$ and thus sum up to the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$.

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].