CriticalLeftToRightMaxima

The entries p that are left to right maxima with exactly p - 1 entries smaller and n - p larger.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000942via ParkingFunction✓↗

CriticalLeftToRightMaxima(parkingFunction)The entries p that are left to right maxima with exactly p - 1 entries smaller and n - p larger.

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 St000942. A left to right maximum is larger than every entry before it.

Definition

Epsil, as the implementation runs it.

on ParkingFunction, over _x

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

Examples

Basic #

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

live ↗

In$\mathrm{CriticalLeftToRightMaxima}({\operatorname{ParkingFunction}}([2, 1, 1, 4, 2]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000942

FindStat's value for [1,1]

live ↗

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

Known value · FindStat St000942

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000942

About

Examples	4
Crosswalk	FindStat