InternalInversions

The pairs i < j with w(i) > w(j) that lie in the same block of the sorted word.

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InternalInversions(word)The pairs i < j with w(i) > w(j) that lie in the same block of the sorted word.

Typed signatures

Type	Package	Applies
(binary_word) -> number	combinatorics	always

Domain: Binary word statistics

Details

• Defined over BinaryWord as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a BinaryWord 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.
• FindStat's St000921. The sorted word is the zeros then the ones, so a pair is internal when both positions are among the first z or both after.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 2 else Apply((z: integer) => Max(Sum(Map(
                                                            Function(Max(Sum(Map(
                                                                                  Function(1 if _x[i] > _x[j] &&
                                                                                                (j <= z ||
                                                                                                 i > z) else 0, j),
                                                                                  Range(i + 1, Length(_x), 1),
                                                                                )), 0), i),
                                                            Range(
                                                                   1,
                                                                   Length(_x) -
                                                                   1,
                                                                 ),
                                                          )), 0), Length(_x) -
                                                                 Max(Sum(_x), 0))

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{InternalInversions}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$4$

Properties #

FindStat's value for 0

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0]), \text{InternalInversions})$Out$0$

Known value · FindStat St000921

FindStat's value for 1

live ↗

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

Known value · FindStat St000921

FindStat's value for 00

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0, 0]), \text{InternalInversions})$Out$0$

Known value · FindStat St000921

About

Examples	4
Crosswalk	FindStat