edit this statistic or download as text // json
Identifier
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