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

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

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

The number of maximal antichains in a poset.

The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.

The largest size of an interval in a poset.

The rank of the poset.

The number of [[/StandardTableaux|standard Young tableaux]] of the partition.