DescentsPlusRecoils

Descents plus recoils: the two counts added, not the union of the indices.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000824via Permutation✓↗

DescentsPlusRecoils(p)Descents plus recoils: the two counts added, not the union of the indices.

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 Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(_x[i] > _x[i + 1], i),
                                     )) + Count(Filter(
                                                       Range(1, Length(_x) - 1),
                                                       Function(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)),
                                                                       )), i),
                                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000824

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000824

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000824

About

Examples	5
Crosswalk	FindStat