InversionsOfThirdEntry

Positions after the third with a smaller entry than p(3).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001556via Permutation✓↗

InversionsOfThirdEntry(p)Positions after the third with a smaller entry than p(3).

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) < 4 else Count(Filter(Range(4, Length(_x)), Function(_x[j] <
                                                                     _x[3], j)))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

In$\mathrm{InversionsOfThirdEntry}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001556

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001556

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

live ↗

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

Known value · FindStat St001556

About

Examples	5
Crosswalk	FindStat