The Catalan number of an irreducible finite Cartan type. The Catalan number of an irreducible finite Cartan type is defined as the product $$ Cat(W) = \prod_{i=1}^n \frac{d_i+h}{d_i}$$ where *$W$ is the Weyl group of the given Cartan type, * $n$ is the rank of $W$, * $d_1 \leq d_2 \leq \ldots \leq d_n$ are the degrees of the fundamental invariants of $W$, and * $h = d_n$ is the corresponding Coxeter number. The Catalan number $Cat(W)$ counts various combinatorial objects, among which are * noncrossing partitions inside $W$, * antichains in the root poset, * regions within the fundamental chamber in the Shi arrangement, * dimensions of several modules in the context of the '''diagonal coininvariant ring''' and of '''rational Cherednik algebras'''. For a detailed treatment and further references, see [1].

The number of antichains in a poset. An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable. An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.

The sum of the parts of an integer partition that are at least two.

The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.

The number of independent sets of vertices of a graph. An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent. This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers. The size of the largest independent set, also called independence number of $G$, is [[St000093]]

The size of a partition. This statistic is the constant statistic of the level sets.

The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

The number of different non-empty partial sums of an integer partition.

The number of edges of a graph.

The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.