CommutativePositiveRootCount

The commutative positive roots: those whose upper ideal in the root poset is abelian.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001173via CartanType✓↗

CommutativePositiveRootCount(cartanType)The commutative positive roots: those whose upper ideal in the root poset is abelian.

Typed signatures

Type	Package	Applies
(cartan_type) -> number	combinatorics	always

Domain: Cartan type statistics

Details

• Defined over CartanType as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a CartanType and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on CartanType, over _x

_x[2] * (_x[2] + 1) / 2 if _x[1] == "A" else _x[2] * (_x[2] + 1) / 2 if _x[1] ==
                                                                        "B" else _x[2] *
                                                                              (_x[2] +
                                                                               1) /
                                                                              2 if _x[1] == "C" else _x[2] *
                                                                                                    (_x[2] + 1) /
                                                                                                    2 +
                                                                                                    (_x[2] - 3) if _x[1] == "D" else (25 if _x[2] == 6 else 34 if _x[2] == 7 else 44) if _x[1] == "E" else 10 if _x[1] == "F" else 3

Examples

Basic #

the type $D_5$

live ↗

In$\mathrm{CommutativePositiveRootCount}(\operatorname{CartanType}((\text{D},5)))$Out$17$

the type $E_6$

live ↗

In$\mathrm{CommutativePositiveRootCount}(\operatorname{CartanType}((\text{E},6)))$Out$25$

Properties #

FindStat's value for ['A',1]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",1)), \text{CommutativePositiveRootCount})$Out$1$

Known value · FindStat St001173

FindStat's value for ['A',2]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",2)), \text{CommutativePositiveRootCount})$Out$3$

Known value · FindStat St001173

FindStat's value for ['A',3]

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{CartanType}(("A",3)), \text{CommutativePositiveRootCount})$Out$6$

Known value · FindStat St001173

About

Examples	5
Crosswalk	FindStat