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 almost positive roots of a finite Cartan type. A root in the root system of a Cartan type is almost positive if it is either positive or simple negative. These are known to be in bijection with cluster variables in the cluster algebra of the given Cartan type, see [1]. This is also equal to the sum of the degrees of the fundamental invariants of the group.

The permanent of the Cartan matrix of a finite Cartan type.

The number of parabolic subgroups of the associated Weyl group. Let $W$ be a Weyl group with simple generators $\mathcal{S} \subseteq W$. A subgroup of $W$ generated by a subset $X \subseteq \mathcal{S}$ is called standard parabolic subgroup. A parabolic subgroup is a subgroup of $W$ that is conjugate to a standard parabolic subgroup. These numbers are called '''parabolic Bell numbers''' and were calculated in [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 number of facets in the order polytope of this poset.

The number of facets in the chain polytope of the poset.

The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.

The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.

The number of non-empty connected induced subgraphs of a graph. More precisely, this is the number of non-empty subsets of the set of vertices of a graph, such that the induced subgraph is connected.