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Identifier
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 [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