**Identifier**

Identifier

Values

['A',1]
=>
1

['A',2]
=>
2

['B',2]
=>
3

['G',2]
=>
5

['A',3]
=>
5

['B',3]
=>
10

['C',3]
=>
10

['A',4]
=>
14

['B',4]
=>
35

['C',4]
=>
35

['D',4]
=>
20

['F',4]
=>
66

['A',5]
=>
42

['B',5]
=>
126

['C',5]
=>
126

['D',5]
=>
77

['A',6]
=>
132

['B',6]
=>
462

['C',6]
=>
462

['D',6]
=>
294

['E',6]
=>
418

['A',7]
=>
429

['B',7]
=>
1716

['C',7]
=>
1716

['D',7]
=>
1122

['E',7]
=>
2431

['A',8]
=>
1430

['B',8]
=>
6435

['C',8]
=>
6435

['D',8]
=>
4290

['E',8]
=>
17342

['A',9]
=>
4862

['B',9]
=>
24310

['C',9]
=>
24310

['D',9]
=>
16445

['A',10]
=>
16796

['B',10]
=>
92378

['C',10]
=>
92378

['D',10]
=>
63206

Description

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 [Wiki], 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 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 [Wiki], 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].

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(positive=True)

Created

Jun 24, 2013 at 21:32 by

**Christian Stump**Updated

Nov 21, 2017 at 09:22 by

**Christian Stump**searching the database

Sorry, this statistic was not found in the database

or

add this statistic to the database – it's very simple and we need your support!