Identifier

Values

=>

Cc0022;cc-rep

['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

search for individual values

searching the database for the individual values of this statistic

/
search for generating function
searching the database for statistics with the same generating function

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

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!