ReflectionLength

n minus the number of cycles — the minimum number of transpositions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000216via Permutation✓↗

ReflectionLength(p)n minus the number of cycles — the minimum number of transpositions.

SageMathenumeratio_stat("Permutation", p, "ReflectionLength")

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) < 1 else Length(_x) - Count(Filter(
                                                    Range(1, Length(_x), 1),
                                                    Function(Fold(
                                                                   Function(
                                                                             w if w <= 0 else 0 if w == i else -1 if w < i else _x[w],
                                                                             w,
                                                                             k,
                                                                           ),
                                                                   _x[i],
                                                                   Range(
                                                                          1,
                                                                          Length(_x),
                                                                          1,
                                                                        ),
                                                                 ) == 0, i),
                                                  ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000216

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000216

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

live ↗

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

Known value · FindStat St000216

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts