Identifier
Identifier
Values
['A',9] generating graphics... => 45
['B',9] generating graphics... => 45
['C',9] generating graphics... => 45
['D',9] generating graphics... => 51
['A',10] generating graphics... => 55
['B',10] generating graphics... => 55
['C',10] generating graphics... => 55
['D',10] generating graphics... => 62
['A',1] generating graphics... => 1
['A',2] generating graphics... => 3
['B',2] generating graphics... => 3
['G',2] generating graphics... => 3
['A',3] generating graphics... => 6
['B',3] generating graphics... => 6
['C',3] generating graphics... => 6
['A',4] generating graphics... => 10
['B',4] generating graphics... => 10
['C',4] generating graphics... => 10
['D',4] generating graphics... => 11
['F',4] generating graphics... => 10
['A',5] generating graphics... => 15
['B',5] generating graphics... => 15
['C',5] generating graphics... => 15
['D',5] generating graphics... => 17
['A',6] generating graphics... => 21
['B',6] generating graphics... => 21
['C',6] generating graphics... => 21
['D',6] generating graphics... => 24
['E',6] generating graphics... => 25
['A',7] generating graphics... => 28
['B',7] generating graphics... => 28
['C',7] generating graphics... => 28
['D',7] generating graphics... => 32
['E',7] generating graphics... => 34
['A',8] generating graphics... => 36
['B',8] generating graphics... => 36
['C',8] generating graphics... => 36
['D',8] generating graphics... => 41
['E',8] generating graphics... => 44
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Description
The number of commutative positive roots in the root system of the given finite Cartan type.
An upper ideal $I$ in the root poset $\Phi^+$ is called abelian if $\alpha,\beta \in I$ implies that $\alpha+\beta \notin \Phi^+$. A positive root is called commutative if the upper ideal it generates is abelian.
The numbers are then given in [1, Theorem 4.4].
References
[1] Panyushev, D. I. The poset of positive roots and its relatives MathSciNet:2218851 arXiv:math/0502385
Code
def statistic(cartan_type):
    n = cartan_type.rank()
    if cartan_type.letter in ["A","B","C","F","G"]:
        n1,n2,n3 = 0,0,0
    elif cartan_type.letter == "D":
        n1,n2,n3 = 1,1,n-3
    elif cartan_type.letter == "E":
        n1,n2,n3 = 1,2,n-4
    return binomial(n+1,2) + n1*n2*n3

Created
May 02, 2018 at 16:59 by Christian Stump
Updated
May 02, 2018 at 16:59 by Christian Stump