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 diagonal coininvariant ring and of rational Cherednik algebras.
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()

Created
Jun 23, 2013 at 12:31 by Christian Stump
Updated
Nov 21, 2017 at 09:21 by Christian Stump