InitialIncreasingSegmentLength

The length of the initial strictly increasing segment.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001904via ParkingFunction✓↗

InitialIncreasingSegmentLength(parkingFunction)The length of the initial strictly increasing segment.

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 St001904. The first i + 1 entries increase exactly when none of the first i neighboring pairs fails to rise.

Definition

Epsil, as the implementation runs it.

on ParkingFunction, over _x

Length(_x) if Length(_x) < 2 else 1 + Max(Sum(Map(
                                                   Function(1 if Max(Sum(Map(
                                                                              Function(1 if j <= i &&
                                                                                            _x[j] >=
                                                                                            _x[j + 1] else 0, j),
                                                                              Range(1, Length(_x) - 1),
                                                                            )), 0) ==
                                                                 0 else 0, i),
                                                   Range(1, Length(_x) - 1),
                                                 )), 0)

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001904

FindStat's value for [1,1]

live ↗

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

Known value · FindStat St001904

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001904

About

Examples	4
Crosswalk	FindStat