InverseMajorIndex

The major index of the inverse: the sum of the values v with v + 1 to the left of v.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000305via Permutation✓↗

InverseMajorIndex(p)The major index of the inverse: the sum of the values v with v + 1 to the left of v.

SageMathenumeratio_stat("Permutation", p, "InverseMajorIndex")

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation 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 Permutation, over _x

0 if Length(_x) < 2 else Sum(Map(
                                  Function(i if Sum(Map(Function(
                                                                  j if _x[j] ==
                                                                       i else 0,
                                                                  j,
                                                                ), Range(
                                                                         1,
                                                                         Length(_x),
                                                                       ))) >
                                                Sum(Map(Function(
                                                                  j if _x[j] ==
                                                                       i + 1 else 0,
                                                                  j,
                                                                ), Range(
                                                                         1,
                                                                         Length(_x),
                                                                       ))) else 0, i),
                                  Range(1, Length(_x) - 1),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{InverseMajorIndex}({\operatorname{Permutation}}([3, 1, 2]))$Out$2$

the one-line word $2413$

live ↗

In$\mathrm{InverseMajorIndex}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$4$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000305

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000305

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{InverseMajorIndex})$Out$1$

Known value · FindStat St000305

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts