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
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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