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Statistic identifier: St000140

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Collection: Finite Cartan types

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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].

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References: [1]   Armstrong, D. Generalized noncrossing partitions and combinatorics of Coxeter groups [[MathSciNet:2561274]] [[arXiv:math/0611106]]
[2]   [[wikipedia:Complex reflection group]]

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Code:
def statistic(ct):
    return ReflectionGroup(ct).catalan_number(positive=True)


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Statistic values:

['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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Created: Jun 24, 2013 at 21:32 by Christian Stump

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Last Updated: Oct 30, 2024 at 17:22 by Martin Rubey