TotalParkingDisplacement

The sum, over the cars, of the spots they drive past: the sum of i - p(i).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000188via ParkingFunction✓↗

TotalParkingDisplacement(parkingFunction)The sum, over the cars, of the spots they drive past: the sum of i - p(i).

Typed signatures

Type	Package	Applies
(parking_function) -> number	combinatorics	always

Domain: Parking function statistics

Details

• Defined over ParkingFunction as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a ParkingFunction 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.
• FindStat's St000188, which is also the area of the Dyck path of the parking function. Each car's displacement is its spot less its preference and the spots are 1 .. n, so the total is n (n + 1) / 2 less the preferences' sum.

Definition

Epsil, as the implementation runs it.

on ParkingFunction, over _x

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

Examples

Basic #

the parking function $[2,1,1,4,2]$

live ↗

In$\mathrm{TotalParkingDisplacement}({\operatorname{ParkingFunction}}([2, 1, 1, 4, 2]))$Out$5$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000188

FindStat's value for [1,1]

live ↗

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

Known value · FindStat St000188

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000188

About

Examples	4
Crosswalk	FindStat