Disorder

Scanning cyclically for 1, then 2, ...: the entries passed that are still larger than the one sought.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000446via Permutation✓↗

Disorder(p)Scanning cyclically for 1, then 2, ...: the entries passed that are still larger than the one sought.

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(Apply(
                                                  (a: number) => Apply(
                                                                        (b: number) => Count(Filter(
                                                                                                     Range(1, Length(_x)),
                                                                                                     Function(_x[q] > v &&
                                                                                                              (a < b &&
                                                                                                               q > a &&
                                                                                                               q < b ||
                                                                                                               a > b &&
                                                                                                               (q > a ||
                                                                                                                q < b)), q),
                                                                                                   )),
                                                                        Sum(Map(
                                                                                 Function(j if _x[j] == v else 0, j),
                                                                                 Range(1, Length(_x)),
                                                                               )),
                                                                      ),
                                                  Sum(Map(
                                                           Function(
                                                                     j if _x[j] ==
                                                                          v - 1 else 0,
                                                                     j,
                                                                   ),
                                                           Range(1, Length(_x)),
                                                         )),
                                                ), v),
                                  Range(1, Length(_x) - 1, 1),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000446

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000446

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000446

About

Examples	5
Crosswalk	FindStat