PermutationCosine

The sum of i p(i).

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PermutationCosine(p)The sum of 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.
• The scalar product of p with the identity: n(n+1)(2n+1) / 6 times the cosine of the angle between them.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{PermutationCosine}({\operatorname{Permutation}}([3, 1, 2]))$Out$11$

the one-line word $2413$

live ↗

In$\mathrm{PermutationCosine}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$25$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000342

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{PermutationCosine})$Out$5$

Known value · FindStat St000342

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{PermutationCosine})$Out$4$

Known value · FindStat St000342

About

Examples	5
Crosswalk	FindStat