MaximalParkingDisplacement

The most spots any car drives past its preferred spot.

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MaximalParkingDisplacement(parkingFunction)The most spots any car drives past its preferred spot.

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 St000943.

Definition

Epsil, as the implementation runs it.

on ParkingFunction, over _x

0 if Length(_x) < 1 else Max(Map(Function(Min(Map(
                                                   Function(t if (1 if Max(Map(
                                                                                Function(Max(Sum(Map(
                                                                                                      Function(1 if j <=
                                                                                                                    i - 1 &&
                                                                                                                    _x[j] >= a &&
                                                                                                                    _x[j] <=
                                                                                                                    _x[i] + t else 0, j),
                                                                                                      Range(1, Length(_x)),
                                                                                                    )), 0) -
                                                                                         (_x[i] + t +
                                                                                          1 -
                                                                                          a), a),
                                                                                Range(1, _x[i] + t),
                                                                              )) >=
                                                                       0 else 0) ==
                                                                 0 else Length(_x), t),
                                                   Range(0, Length(_x) - _x[i]),
                                                 )), i), Range(1, Length(_x))))

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000943

FindStat's value for [1,1]

live ↗

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

Known value · FindStat St000943

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000943

About

Examples	4
Crosswalk	FindStat