**Identifier**

Identifier

Values

['A',1]
=>
2

['A',2]
=>
5

['B',2]
=>
6

['G',2]
=>
8

['A',3]
=>
14

['B',3]
=>
20

['C',3]
=>
20

['A',4]
=>
42

['B',4]
=>
70

['C',4]
=>
70

['D',4]
=>
50

['F',4]
=>
105

['A',5]
=>
132

['B',5]
=>
252

['C',5]
=>
252

['D',5]
=>
182

['A',6]
=>
429

['B',6]
=>
924

['C',6]
=>
924

['D',6]
=>
672

['E',6]
=>
833

['A',7]
=>
1430

['B',7]
=>
3432

['C',7]
=>
3432

['D',7]
=>
2508

['E',7]
=>
4160

['A',8]
=>
4862

['B',8]
=>
12870

['C',8]
=>
12870

['D',8]
=>
9438

['E',8]
=>
25080

['A',9]
=>
16796

['B',9]
=>
48620

['C',9]
=>
48620

['D',9]
=>
35750

['A',10]
=>
58786

['B',10]
=>
184756

['C',10]
=>
184756

['D',10]
=>
136136

Description

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

For a detailed treatment and further references, see [1].

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].

References

[1]

[2] wikipedia:Complex reflection group

**Armstrong, D.***Generalized noncrossing partitions and combinatorics of Coxeter groups*MathSciNet:2561274 arXiv:math/0611106[2] wikipedia:Complex reflection group

Code

def statistic(ct): return ReflectionGroup(ct).catalan_number()

Created

Jun 23, 2013 at 12:31 by

**Christian Stump**Updated

Nov 21, 2017 at 09:21 by

**Christian Stump**searching the database

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