DescentWeightMinusInversions

The sum of i(n - i) over the descents i, less the inversions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001911via Permutation✓↗

DescentWeightMinusInversions(p)The sum of i(n - i) over the descents i, less the inversions.

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

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{DescentWeightMinusInversions}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001911

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001911

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001911

About

Examples	5
Crosswalk	FindStat