DescentDifferenceSum

The sum of p(i) - p(i+1) over the descents i.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000030via Permutation✓↗

DescentDifferenceSum(p)The sum of p(i) - p(i+1) over the descents i.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.
• Also called the drop of a permutation.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{DescentDifferenceSum}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$2$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000030

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000030

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000030

About

Examples	6
Crosswalk	FindStat