NormalizedSumOfDistancesToGreaterElement

The same sum, each gap one less: the entries strictly between a position and its nearest greater entry.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000624via Permutation✓↗

NormalizedSumOfDistancesToGreaterElement(p)The same sum, each gap one less: the entries strictly between a position and its nearest greater entry.

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) < 1 else Sum(Map(Function(Min(Join(
                                                    [i, Length(_x) + 1 - i],
                                                    Map(
                                                         Function(Abs(j - i), j),
                                                         Filter(
                                                                 Range(
                                                                        1,
                                                                        Length(_x),
                                                                      ),
                                                                 Function(_x[j] >
                                                                          _x[i], j),
                                                               ),
                                                       ),
                                                  )) - 1, i), Range(
                                                                    1,
                                                                    Length(_x),
                                                                  )))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000624

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000624

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

live ↗

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

Known value · FindStat St000624

About

Examples	5
Crosswalk	FindStat