- FindStat

Identifier

St001173: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['A',1]=>1
['A',2]=>3
['B',2]=>3
['G',2]=>3
['A',3]=>6
['B',3]=>6
['C',3]=>6
['A',4]=>10
['B',4]=>10
['C',4]=>10
['D',4]=>11
['F',4]=>10
['A',5]=>15
['B',5]=>15
['C',5]=>15
['D',5]=>17
['A',6]=>21
['B',6]=>21
['C',6]=>21
['D',6]=>24
['E',6]=>25
['A',7]=>28
['B',7]=>28
['C',7]=>28
['D',7]=>32
['E',7]=>34
['A',8]=>36
['B',8]=>36
['C',8]=>36
['D',8]=>41
['E',8]=>44
['A',9]=>45
['B',9]=>45
['C',9]=>45
['D',9]=>51
['A',10]=>55
['B',10]=>55
['C',10]=>55
['D',10]=>62
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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