Identifier
Identifier
Values
['A',1] generating graphics... => 1
['A',2] generating graphics... => 2
['B',2] generating graphics... => 3
['G',2] generating graphics... => 5
['A',3] generating graphics... => 5
['B',3] generating graphics... => 10
['C',3] generating graphics... => 10
['A',4] generating graphics... => 14
['B',4] generating graphics... => 35
['C',4] generating graphics... => 35
['D',4] generating graphics... => 20
['F',4] generating graphics... => 66
['A',5] generating graphics... => 42
['B',5] generating graphics... => 126
['C',5] generating graphics... => 126
['D',5] generating graphics... => 77
['A',6] generating graphics... => 132
['B',6] generating graphics... => 462
['C',6] generating graphics... => 462
['D',6] generating graphics... => 294
['E',6] generating graphics... => 418
['A',7] generating graphics... => 429
['B',7] generating graphics... => 1716
['C',7] generating graphics... => 1716
['D',7] generating graphics... => 1122
['E',7] generating graphics... => 2431
['A',8] generating graphics... => 1430
['B',8] generating graphics... => 6435
['C',8] generating graphics... => 6435
['D',8] generating graphics... => 4290
['E',8] generating graphics... => 17342
['A',9] generating graphics... => 4862
['B',9] generating graphics... => 24310
['C',9] generating graphics... => 24310
['D',9] generating graphics... => 16445
['A',10] generating graphics... => 16796
['B',10] generating graphics... => 92378
['C',10] generating graphics... => 92378
['D',10] generating graphics... => 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 [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] 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