VisibleInversions

Pairs i < j with p(j) <= min(i, p(i)).

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VisibleInversions(p)Pairs i < j with p(j) <= min(i, p(i)).

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(Sum(Map(
                                                   Function(1, j),
                                                   Filter(
                                                           Range(
                                                                  i + 1,
                                                                  Length(_x),
                                                                ),
                                                           Function(_x[j] <= i &&
                                                                    _x[j] <=
                                                                    _x[i], j),
                                                         ),
                                                 )), i), Range(1, Length(_x) - 1)))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001726

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001726

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001726

About

Examples	5
Crosswalk	FindStat