ReducedReflectionLength

The total displacement less the inversions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000809via Permutation✓↗

ReducedReflectionLength(p)The total displacement 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.
• Twice the depth less the length.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Sum(Map(
                                  Function(Abs(_x[i] - i), i),
                                  Range(1, Length(_x)),
                                )) - 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$\mathrm{ReducedReflectionLength}({\operatorname{Permutation}}([3, 1, 2]))$Out$2$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000809

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000809

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

live ↗

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

Known value · FindStat St000809

About

Examples	5
Crosswalk	FindStat