The positive Catalan number of an irreducible finite Cartan type. The positive Catalan number of an irreducible finite Cartan type is defined as the product $$Cat^+(W) = \prod_{i=1}^n \frac{d_i-2+h}{d_i} = \prod_{i=1}^n \frac{d^*_i+h}{d_i}$$ 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 $W$, * $d^*_1 \geq d^*_2 \geq \ldots \geq d^*_n$ are the codegrees for $W$, see [2], and * $h = d_n$ is the corresponding Coxeter number. The positive Catalan number $Cat^+(W)$ counts various combinatorial objects, among which are * noncrossing partitions of full Coxeter support inside $W$, * antichains not containing simple roots in the root poset, * bounded regions within the fundamental chamber in the Shi arrangement. For a detailed treatment and further references, see [1].

The Coxeter number of a finite Cartan type. The Coxeter number $h$ for the Weyl group $W$ of the given finite Cartan type is defined as the order of the product of the Coxeter generators of $W$. Equivalently, this is equal to the maximal degree of a fundamental invariant of $W$, see also [[St000138]].

The number of full-support reflections in the Weyl group of a finite Cartan type. A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.

The number of reflections of the Weyl group of a finite Cartan type. By the one-to-one correspondence between reflections and reflecting hyperplanes, this is also the number of reflecting hyperplanes. This is given by $nh/2$ where $n$ is the rank and $h$ is the Coxeter number.

The number of factorizations of any Coxeter element into reflections of a finite Cartan type. The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type. This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6]. As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element $$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$ in type $A_2$.

The maximal order of an element in the Weyl group of a given Cartan type. For the symmetric group, this is [[oeis:A000793]]

The number of minimal subsets of reflections that generate the group. $~$ This counts the number of minimal subsets $S \subseteq R$ of reflections $R \subseteq W$ that generate the group $W$.

The minimal degree of a faithful permutation representation of a Weyl group. Data are from [1, Table 1].

The number of non-isomorphic subgroups of the Weyl group of a finite Cartan type. This statistic returns the number of non-isomorphic abstract groups. See [[St001155]] for the number of conjugacy classes of subgroups.

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].