- FindStat

Identifier

St000140: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['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
                    

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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 [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].

References

[1] 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

Oct 30, 2024 at 17:22 by Martin Rubey