FlagMajorIndex

The number of negative entries plus twice the sum of the descents, a negative first entry counting as a descent at 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001433via SignedPermutation✓↗

FlagMajorIndex(p)The number of negative entries plus twice the sum of the descents, a negative first entry counting as a descent at 0.

Typed signatures

Type	Package	Applies
(signed_permutation) -> number	combinatorics	always

Domain: Signed permutation statistics

Details

• Defined over SignedPermutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SignedPermutation 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.
• The descent at 0 adds 0 to the sum, so only the descents between entries weigh.

Definition

Epsil, as the implementation runs it.

on SignedPermutation, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] < 0, i),
                                     )) + (0 if Length(_x) < 2 else Max(2 *
                                                                       Sum(Map(
                                                                                Function(i if _x[i] >
                                                                                              _x[i + 1] else 0, i),
                                                                                Range(1, Length(_x) - 1),
                                                                              )), 0))

Examples

Basic #

the signed word $2\,\bar{1}\,\bar{3}$

live ↗

In$\mathrm{FlagMajorIndex}({\operatorname{SignedPermutation}}([2, -1, -3]))$Out$8$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([1]), \text{FlagMajorIndex})$Out$0$

Known value · FindStat St001433

FindStat's value for [-1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([-1]), \text{FlagMajorIndex})$Out$1$

Known value · FindStat St001433

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([1, 2]), \text{FlagMajorIndex})$Out$0$

Known value · FindStat St001433

About

Examples	4
Crosswalk	FindStat