TotalSquaredDisplacement

The sum of (p(i) - i)^2.

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TotalSquaredDisplacement(p)The sum of (p(i) - i)^2.

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 InversionSum, though neither definition shows it. FindStat calls it Spearman's rho of p against the identity.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Sum(Map(
                                  Function((_x[i] - i) ^ 2, i),
                                  Range(1, Length(_x)),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{TotalSquaredDisplacement}({\operatorname{Permutation}}([3, 1, 2]))$Out$6$

the one-line word $2413$

live ↗

In$\mathrm{TotalSquaredDisplacement}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$10$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000828

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000828

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000828

About

Examples	5
Crosswalk	FindStat