ScaledReciprocalPositionValueSum

The sum of 1/(i + p(i)) times the least common multiple of 1 .. 2n, an integer.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001293via Permutation✓↗

ScaledReciprocalPositionValueSum(p)The sum of 1/(i + p(i)) times the least common multiple of 1 .. 2n, an integer.

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.
• Each term is an integer, since i + p(i) is at most 2n and divides the scale (St001293).

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Apply(
                                (scale: integer) => Max(Sum(Map(
                                                                 Function(scale /
                                                                          (i +
                                                                           _x[i]), i),
                                                                 Range(
                                                                        1,
                                                                        Length(_x),
                                                                      ),
                                                               )), 0),
                                Fold(
                                      Function(LCM(a, b), a, b),
                                      1,
                                      Range(1, 2 * Length(_x), 1),
                                    ),
                              )

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{ScaledReciprocalPositionValueSum}({\operatorname{Permutation}}([3, 1, 2]))$Out$47$

the one-line word $2413$

live ↗

In$\mathrm{ScaledReciprocalPositionValueSum}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$750$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001293

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{ScaledReciprocalPositionValueSum})$Out$9$

Known value · FindStat St001293

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{ScaledReciprocalPositionValueSum})$Out$8$

Known value · FindStat St001293

About

Examples	5
Crosswalk	FindStat